volume polynomials and duality algebras of...

Arnold Math J.DOI 10.1007/s40598-016-0048-4

RESEARCH CONTRIBUTION

Volume Polynomials and Duality Algebrasof Multi-Fans

Anton Ayzenberg1 · Mikiya Masuda2

Received: 17 October 2015 / Revised: 12 November 2015 / Accepted: 23 June 2016© Institute for Mathematical Sciences (IMS), Stony Brook University, NY 2016

Abstract We introduce a theory of volume polynomials and corresponding dualityalgebras of multi-fans. Any complete simplicial multi-fan Δ determines a volumepolynomial VΔ whose values are the volumes of multi-polytopes based on Δ. Thishomogeneous polynomial is further used to construct a Poincare duality algebraA∗(Δ). We study the structure and properties of VΔ andA∗(Δ) and give applicationsand connections to other subjects, such as Macaulay duality, Novik–Swartz theory offace rings of simplicial manifolds, generalizations of Minkowski’s theorem on con-vex polytopes, cohomology of torus manifolds, computations of volumes, and linearrelations on the powers of linear forms. In particular, we prove that the analogue ofthe g-theorem does not hold for multi-polytopes.

Keywords Multi-fan · Multi-polytope · Volume polynomial · Poincare dualityalgebra · Macaulay duality · Stanley–Reisner ring · Minkowski theorem ·Minkowski relations · Torus manifold

Mathematics Subject Classification 52A39 · 52B11 · 05E45 · 52C35 · 05E40 ·13H10 · 52B05 · 52B40 · 52B70 · 57N65 · 55N91 · 28A75 · 51M25 · 13A02

B Anton [email protected]; [email protected]

1 Faculty of Mathematics, National Research University Higher School of Economics,Moscow, Russia

2 Department of Mathematics, Osaka City University, Osaka, Japan

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http://crossmark.crossref.org/dialog/?doi=10.1007/s40598-016-0048-4&domain=pdf

http://orcid.org/0000-0003-4037-9249

A. Ayzenberg, M. Masuda

1 Introduction

There is a fundamental correspondence in algebraic geometry (Fulton 1993):

{Toric varieties

of complex dimension n

}�

{Rational fans in R

n} .

One can read the information about toric variety from its fan. Complete toric vari-eties correspond to complete fans, non-singular varieties correspond to non-singularfans, and projective toric varieties correspond to normal fans of convex polytopes.Combinatorics of a fan and geometry of a toric variety are closely connected. Inparticular, the rays of a fan correspond to the divisors on toric variety and higherdimensional cones correspond to the intersections of divisors.

In theworkHattori andMasuda (2003) expanded this setting to topological categoryand generalized the above-mentioned correspondence in the following way:

{Torus manifolds

of real dimension 2n

}�

{Nonsingular multi-fans in R

n} , (1.1)

which will be explained in a minute.Let X be a smooth closed oriented 2n-manifold with an effective action of an

n-dimensional compact torus T and at least one fixed point. A closed, connected,codimension two submanifold of X will be called characteristic if it is a connectedcomponent of the fixed point set of a certain circle subgroup S of T , and if it contains atleast one T -fixed point. The manifold X together with a preferred orientation of eachcharacteristic submanifold is called a torus manifold. Characteristic submanifolds arethe analogues of divisors on a toric variety.

Note, that there is no one-to-one correspondence in (1.1): there may be different(in any sense) torus manifolds producing the same multi-fan. Nevertheless, multi-fansprovide a convenient tool to study such manifolds.

A multi-fan is the central object of this paper. We recall the precise definitionlater. Informally, a multi-fan is a collection of cones in V ∼= R

n with apex at theorigin, coming with multiplicities and satisfying certain restrictions. Sometimes it isconvenient to assume that there is a fixed lattice N ⊂ V , and the rays ofΔ are rationalwith respect to N . The cones of a multi-fan may overlap nontrivially, which makes amulti-fan more general and flexible object than an ordinary fan, and provides manynontrivial examples.

A multi-polytope is defined as follows. Let Δ be a simplicial multi-fan in V ∼= Rn .

For each ray li ∈ Δ, we specify an affine hyperplane Hi ⊂ V ∗ orthogonal to the linearspan of li . A tuple P = (Δ, H1, . . . , Hm) is called a simple multi-polytope based onΔ. The relation of the multi-polytope to the multi-fan on which it is based, is exactlythe same as the relation of a polytope to its normal fan.

For any multi-polytope P ⊂ V ∗ there is a function DHP : V ∗\⋃ Hi → Z (thenotation stands for Duistermaat–Heckman, see Hattori andMasuda 2003). Informally,for a generic point x ∈ V ∗ the valueDHP (x) indicates howmany times the “boundary”

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Volume Polynomials and Duality Algebras of Multi-Fans

of P wraps around x . The precise definition is given in Sect. 3. For an ordinary simpleconvex polytope this function takes value 1 inside the polytope, and 0 outside.

A multi-fan Δ is called complete if it satisfies certain mild conditions (see Hattoriand Masuda 2003 or Definition 2.5 below). For multi-polytopes based on completesimplicial multi-fans, the function DHP is compactly supported. We can define thevolume of a multi-polytope P as an integral

Vol(P) :=∫V ∗

DHP dμ

(the measure μ is chosen such that the volume of a fundamental domain of the duallattice N∗ is 1).

For a given simplicialmulti-fanΔ consider the space Poly(Δ) of allmulti-polytopesbased on Δ. Following Timorin (1999) we call it the space of analogous multi-polytopes. To specify an affine hyperplane orthogonal to a line 〈li 〉 ⊂ V one needs asingle real number ci , the normalized distance from Hi to the origin taken with sign.This number is called the support parameter. Thus the space Poly(Δ) is isomorphicto R

m , where m is the number of rays of Δ. Support parameters (c1, . . . , cm) providethe canonical coordinates on Poly(Δ).

If Δ is complete, the volume gives a function on the space of analogous polytopes:Poly(Δ) → R, P → Vol(P). Similarly to the case of actual convex polytopes,studied by Pukhlikov–Khovanskii (Khovanskii and Pukhlikov 1992a), this function isa homogeneous polynomial in the support parameters.

Theorem 1.1 (Hattori and Masuda 2003) Let Δ be a complete simplicial multi-fanin R

n with m rays. There exists a homogeneous polynomial VΔ ∈ R[c1, . . . , cm] ofdegree n such that VΔ(c1, . . . , cm) = Vol(P) for a multi-polytope P ∈ Poly(Δ) withsupport parameters (c1, . . . , cm).

Following the approach introduced in Khovanskii and Pukhlikov (1992b) anddeveloped in Timorin (1999), we proceed as follows. Consider the ring D of dif-ferential operators with constant coefficients, acting on R[c1, . . . , cm]. We have D =R[∂1, . . . , ∂m], where ∂i = ∂

∂ci. It is convenient to double the degree, so we assume

that deg ∂i = 2. Given any nonzero homogeneous polynomial Ψ ∈ R[c1, . . . , cm] ofdegree n, consider the subspace Ann(Ψ ) ⊂ D, Ann(Ψ ) = {D ∈ D | DΨ = 0}. Itis easily seen, that Ann(Ψ ) is a graded ideal, and the quotient algebra D/Ann(Ψ )

is finite-dimensional and vanishes in degrees >2n. Moreover, D/Ann(Ψ ) is acommutative Poincare duality algebra of formal dimension 2n (see e.g. Timorin1999, Prop. 2.5.1).

Now consider a complete simplicial multi-fan Δ and apply this construction tothe volume polynomial VΔ. In result we obtain a Poincare duality algebra A∗(Δ) :=D/Ann(VΔ) associated with a multi-fan Δ. The main goal of this work is to study thevolume polynomials and investigate the structure of the corresponding algebras andto show their relation to other topics in combinatorics, convex geometry, commutativealgebra, and topology.

The work has the following structure. In Sects. 2 and 3 we review the basic notionsof the theory of multi-fans and in Sect. 4 we review the notion of the index map which

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is the key ingredient in the construction of the volume polynomial. In the work Hattoriand Masuda (2003), introducing multi-fans, the existence of a lattice N ∼= Z

n ⊂ Vwas assumed, so that multi-fans are non-singular (or at least rational) with respectto this lattice. In our paper we consider general multi-fans, probably non-rational.Instead of a lattice we assume that the ambient space V has a fixed inner product. Thisallows, in particular, to define and compute volumes of multi-polytopes in V ∗ = Vof dimensions smaller than n (dealing with lattices, only unimodular volumes makesense). The exposition of the multi-fan theory is built to comply with this continuoussetting. Nevertheless, all statements in the introductory sections follow from theirlattice analogues discussed in Hattori and Masuda (2003).

In Sect. 5 we prove the basic enumerative properties of the volume polynomial.While the values of VΔ are the volumes of multi-polytopes, the values of its partialderivatives are the volumes of proper faces of these multi-polytopes up to certainconstants. These relations will be used further in Sect. 9.

In Sect. 6 we prove a general formula (actually, a family of formulas) for thevolume polynomial, and indicate a geometrical procedure which allows to find non-trivial linear identities on the powers of linear forms. For actual convex polytopesour formula coincides with the Lawrence’s formula (Lawrence 1991), which is wellknown in computational geometry.

In Sect. 7 we review the general correspondence between homogeneous poly-nomials and Poincare duality algebras, known as the Macaulay duality. Using thiscorrespondence we obtain an algebraA∗(Δ) as a Poincare duality algebra correspond-ing to the volume polynomial VΔ. One way to obtain this algebra is via differentialoperators as discussed above. Another way involves the index map of a multi-fan.

The structure of multi-fan algebras in some particular cases is described in Sect. 8.Every (complete simplicial) multi-fan has an underlying simplicial cycle. If this cycleis a homology sphere K , thenA∗(Δ) is the quotient of the Stanley–Reisner algebra ofK by a linear system of parameters, and the dimensions of its graded components arethe h-numbers of K . This is similar to ordinary fans. If the underlying simplicial cycleis a homology manifold, the algebra A∗(Δ) is the quotient of the Stanley–Reisneralgebra by the linear system of parameters and by the certain ideal introduced andstudied by Novik and Swartz (2009a, b). In this case the dimensions of the gradedcomponents of A∗(Δ) are the h′′-numbers of K . A short exposition of the Novik–Swartz theory is provided.

Section 9 aims to generalize a classical Minkowski theorem on convex polytopes tomulti-polytopes. The direct Minkowski theorem has a straightforward generalizationwhich can be used to obtain linear relations in the algebra A∗(Δ). On the other hand,the inverse Minkowski theorem, properly formulated, is controlled by the power mapA2(Δ) → A2n−2(Δ), a → an−1.

In Sect. 10 we answer the question which polynomials are volume polynomials ofmulti-fans, and which Poincare duality algebras are algebras of multi-fans. We provethat every Poincare duality algebra generated in degree 2 is isomorphic to A∗(Δ) forsome complete simplicial multi-fanΔ. This proves that the analogue of the g-theoremfails for multi-fans.

The basic operations on multi-fans, such as flips and connected sums, and theireffects to multi-fan algebras are described in Sect. 11. In particular, we prove that,

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under flips, the dimensions of graded components of A∗(Δ) change similarly to h-numbers of simplicial complexes.

Finally, in Sect. 12 we discuss the relation of A∗(Δ) to the cohomology of torusmanifolds. It is known that, for complete smooth toric variety X , the cohomology ringH∗(X; R) coincides with the algebraA∗(ΔX ) of the corresponding fan. Situationwithgeneral torus manifolds and their multi-fans is more complicated. Nevertheless, in acertain sense, themulti-fan algebraA∗(ΔX ) gives a “lower bound” for the cohomologyof a torus manifold X .

2 Definitions: Multi-Fans

2.1 Multi-Fans as Parametrized Collections of Cones

Let us recall the definition and basic properties of multi-fans. This exposition followsthe lines of Hattori and Masuda (2003).

Consider an oriented vector space V ∼= Rn with a lattice N ⊂ V , N ∼= Z

n . Asubset of the form κ = {r1v1 +· · ·+ rkvk | ri � 0} for given v1, . . . , vk ∈ V is calleda cone in V . Dimension of κ is the dimension of the linear hull of κ . A cone is calledstrongly convex if it contains no line through the origin. In the following all cones areassumed strongly convex.

Using classical construction of supporting hyperplane one can define the faces ofκ , which are also the cones of smaller dimensions. If the generating set v1, . . . , vkmay be chosen linearly independent (resp. rational, the part of basis of the lattice N ),κ is called simplicial (resp. rational, unimodular). Let Cone(V ) denote the set of allcones in V . This set obtains a partial order: κ1 ≺ κ2 whenever κ1 is a face of κ2.

Let Σ be a finite partially ordered set with the minimal element ∗. Suppose thereis a map C : Σ → Cone(V ) such that

1. C(∗) = {0};2. If I < J for I, J ∈ Σ , then C(I ) ≺ C(J );3. For any J ∈ Σ the map C restricted on {I ∈ Σ | I � J } is an isomorphism of

ordered sets onto {κ ∈ Cone(V ) | κ � C(J )}.The image C(Σ) is a finite set of cones in V . We may think of a pair (Σ,C) as a

set of cones in V labeled by the ordered set Σ .The poset Σ obtains a rank function: rk(I ) := dimC(I ). The set of elements in Σ

having maximal rank n is denoted Σ 〈n〉.Consider an arbitrary function σ : Σ 〈n〉 → {−1,+1} called a sign function.

Definition 2.1 (Old definition) The triple Δ := (Σ,C, σ ) is called a multi-fan in V .The number n = dim V is called the dimension of Δ.

Multi-fan Δ is called simplicial (resp. rational, non-singular) if the values of C aresimplicial (resp. rational, unimodular) cones. In the following we will always assumethat Δ is simplicial. Then every cone of Δ is simplicial and property (3) of the map Cimplies that Σ is a simplicial poset. Recall that a poset Σ is called simplicial if anylower order ideal Σ�J := {I ∈ Σ | I � J } is isomorphic to the poset of faces of asimplex (i.e. a boolean lattice).

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2.2 Multi-Fans as Pairs of Weight and Characteristic Functions

Note that Definition 2.1 of a multi-fan slightly differs from the definition of multi-fangiven in Hattori and Masuda (2003). To establish the correspondence consider thefollowing construction. Let [m] = {1, . . . ,m} denote the set of vertices of Σ .

The signs of maximal simplices in Σ determine two functions on([m]n

), the set of

all n-subsets of [m]:

w± :([m]

n

)→ Z�0,

where w+({i1, . . . , in}) (resp. w−({i1, . . . , in})) equals the number of simplices I ∈Σ 〈n〉 on the vertices {i1, . . . , in} having sign +1 (resp. −1). Although both functionsw+, w− are important by topological reasons (see Hattori and Masuda 2003), onlytheir difference w := w+ − w− is relevant to our work. So far w is a function whichassigns an integral number to each n-subset of [m]. Let us consider a pure simplicialcomplex K on the set [m] whose maximal simplices K 〈n〉 are the subsets I ⊂ [m]satisfying w(I ) = 0. To reach greater generality we allow w to take real values, thus

w :([m]

n

)→ R.

Each vertex i ∈ [m] corresponds to a ray (i.e. 1-dimensional cone) ofΔ. We choosea generator in each ray. This gives a so called characteristic function λ : [m] → V ,such that the ray C(i) is generated by λ(i) for every i ∈ [m]. It satisfies the followingproperty:

if {i1, . . . , ik} ∈ K , then λ(i1), . . . , λ(ik) ∈ V are linearly independent.

This condition is called ∗-condition.Note that in Hattori andMasuda (2003) all multi-fans were assumed rational. In this

case the generator λ(i) can be chosen canonically as a unique primitive integral vectorcontained in C(i). Since we want to include non-rational simplicial multi-fans in ourconsideration, we should specify the generators somehow in order for the subsequentcalculations to make sense.

Finally we get to the following definition

Definition 2.2 (New definition) A triple (K , w, λ) is called a simplicial multi-fan inV . Here w : ([m]

n

) → R is a weight function, K is a simplicial complex which is thesupport of w, and λ : [m] → V is a characteristic function. Characteristic functionsatisfies ∗-condition with respect to K : if I = {i1, . . . , ik} ∈ K , then the vectorsλ(i1), . . . , λ(ik) are linearly independent in V .

Here K may have ghost vertices, i.e. i ∈ [m] such that {i} /∈ K . The value ofcharacteristic function in such vertices may be arbitrary (even zero). In the followingwe will not pay too much attention to ghost vertices since their presence does notaffect the calculations.

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Strictly speaking, the new definition is not equivalent to the old one, sincewe cannotrestore the posetΣ and the sign function σ : Σ 〈n〉 → {±1}whenw takes non-integralvalues. Even in the integral case we cannot restore Σ uniquely. On the other hand, aswas shown above, every multi-fan in the sense of old definition determines a multi-fanin the sense of new definition. We will work with the new definition most of the time.

Remark 2.3 When passing from the old definition to the new one, we may lose animportant information. For example consider the multi-fan in R

2 = 〈e1, e2〉 whosemaximal cones are two copies of the non-negative cone (i.e. the cone generated bybasis vectors e1, e2), and two rays are generated by e1 and e2. One of the maximalcones is taken with the sign +1 and the other with the sign −1. We remark thatsuch multi-fan corresponds to the torus manifold S4 (Hattori and Masuda 2003). Wehave w+({1, 2}) = w−({1, 2}) = 1, therefore w({1, 2}) = 0. Thus K is empty(equivalently, w : ([m]

n

) → R vanishes).One way to avoid such situations is to assume in the beginning that Σ itself is a

simplicial complex rather than a general simplicial poset. In this case K coincides withΣ and the weight function w on K coincides with the sign function σ . In particular,w takes the value ±1 on each maximal simplex of K (see Example 2.9 below).

2.3 Underlying Simplicial Chain

Let�[m] denote an abstract simplex on the vertex set [m], and let�(n−1)[m] be its (n−1)-

skeleton. Every subset I ⊂ [m], |I | = n may be considered as a maximal simplex of�(n−1)

[m] . If I ∈ K 〈n〉, then we can orient I as follows: we say that the order of vertices(i1, . . . , in) of I is positive if and only if the basis (λ(i1), . . . , λ(in)) determines thepositive orientation of V .

Definition 2.4 The element

wch =∑

I∈K 〈n〉w(I )I ∈ Cn−1(K ; R) ⊆ Cn−1

(�(n−1)

[m] ; R

)

is called the underlying chain of a multi-fan Δ. Here Cn−1(K ; R) denotes the groupof simplicial chains of K .

2.4 Complete Multi-Fans

Let us briefly recall the notion of projected multi-fan. We give the construction interms of new definition of multi-fan although the similar construction may be givenin terms of simplicial posets and sign functions.

Let Δ = (K , w, λ) be a simplicial multi-fan in the space V , and let I ={i1, . . . , ik} ∈ K be a simplex. Let VI denote the quotient vector space V/〈λ(i1), . . . ,λ(ik)〉. Consider the multi-fan ΔI = (lkK I, wI , λI ) in VI defined as follows:

– lkK I := {J ⊂ [m]\I | I ∪ J ∈ K } is the link of the simplex I in K .– wI (J ) := w(I ∪ J ) for every J ∈ lkK I , |J | = n − |I |.

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– λI ( j) is the image of λ( j) ∈ V under the natural projection V → VI =V/〈λ(i1), . . . , λ(ik)〉. It is easily seen that λI satisfies ∗-condition.If we choose some orientation of a simplex I ∈ K , the space VI obtains an ori-

entation induced from V . To be precise, let us say that the basis ([v1], . . . , [vn−k])determines a positive orientation of VI if the basis (v1, . . . , vn−k, λ(i1), . . . , λ(ik)) isa positive basis of V for a chosen positive order (i1, . . . , ik) of vertices of I .

We call ΔI the projected multi-fan of Δ. The construction satisfies the hereditaryrelation (ΔI1)I2 = ΔI1�I2 whenever it makes sense, and there holds Δ∅ = Δ.

Let us call a vector v ∈ V generic with respect to Δ if it is not contained inthe vector subspaces spanned by the cones of Δ of dimensions < n. For any suchv define the number dv = ∑

w(I ) ∈ R, where the sum is taken over all subsetsI = {i1, . . . , in} ⊂ [m] such that the cone generated by λ(i1), . . . , λ(in) contains v.

Definition 2.5 The multi-fan Δ is called pre-complete if dv does not depend on ageneric vector v ∈ V . In this case dv is called the degree of Δ. The multi-fan Δ iscalled complete if the projected multi-fan ΔI is pre-complete for any simplex I ∈ K .

Remark 2.6 Note that this definition allows w to be constantly zero. We call a multi-fan zero if its weight function is constantly zero. A zero multi-fan is pre-complete andtherefore complete.

Proposition 2.7 A multi-fan Δ is complete if and only if its underlying simplicialchain wch ∈ Cn−1(�(n−1)

[m] ; R) is a cycle, that is dwch = 0 for the standard simplicial

differential d : Cn−1(�(n−1)[m] ; R) → Cn−2(�(n−1)

[m] ; R) (if n = 1, we assume that

d : C0(�(n−1)[m] ; R) → R is the augmentation map).

Proof In the case whenw takes only integral values, the statement is proved in Hattoriand Masuda (2003, Sec. 6). If w takes only rational values, scaling the values of w

by a common denominator reduces the task to the integral case. It remains to provethe statement for real-valued w. Both conditions “Δ is complete” and “dwch = 0”determine rational vector subspaces in the space of all possible weight functions (it isnot difficult to define the pre-completeness condition in terms of the “wall-crossingrelations”, which are linear relations on w(I ) with integral coefficients). Thus therational case implies the real case. ��

For convenience we summarize the discussion by the following definition.

Definition 2.8 (Complete simplicial multi-fan) A complete simplicial multi-fan is apair (wch, λ), wherewch = ∑

I⊂[m],|I |=n w(I )I ∈ Zn−1(�(n−1)[m] ) is a simplicial cycle

on m vertices, and λ : [m] → V is any function satisfying the condition: {λ(i)}i∈I isa basis of V if |I | = n and w(I ) = 0.

For a completemulti-fanΔ the correspondinghomology class [wch] ∈ Hn−1(K ; R)

⊂ Hn−1(�(n−1)[m] ; R) will be denoted [Δ] and called the underlying homology class

of Δ. Since Cn(�(n−1)[m] ; R) = 0, the groups Zn−1(�(n−1)

[m] ) and Hn−1(�(n−1)[m] ) may be

identified. Thus wch and [Δ] are just two different notations for the same object.

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Example 2.9 Apseudomanifold is a pure simplicial complex such that each simplex ofcodimension one is contained in exactly two maximal simplices. It is called orientableif all its maximal simplices can be simultaneously oriented so that orientations ofneighboring simplices are compatible. One obvious way to obtain a complete multi-fan is to start with any oriented pseudomanifold K of dimension n − 1 on the set ofvertices [m], and take any characteristic function λ : [m] → V . Since K is oriented,every maximal simplex I of K becomes oriented, but this orientation may be differentfrom the one determined by characteristic function (see Sect. 2.3). Let w(I ) be +1or −1 depending on whether these two orientations agree or not. Let us extend theweight function by zeroes to non-simplices of K . The corresponding simplicial chainwch = ∑

I w(I )I ∈ Cn−1(�(n−1)[m] ; R) is closed, since it is exactly the fundamental

chain of K in �(n−1)[m] . Therefore, (wch, λ) is a complete simplicial fan.

Example 2.10 The previous example may be restricted to the case when K is a homol-ogy sphere or homology manifold. We will study these two cases in more detail inSect. 8.

We say that Δ is based on an orientable simplicial pseudomanifold K if the corre-sponding simplicial cycle is given by K .

There is one interesting feature of (complete) multi-fans revealed byDefinitions 2.2and 2.8. The multi-fans with the given set of vertices [m] and the given characteris-tic function λ form a vector space: we may add them by adding their weights andmultiply by real numbers by scaling their weights. Let MultiFansλ denote the vectorspace of complete multi-fans with the given characteristic function λ. This space maybe identified with certain vector subspace of Zn−1(�(n−1)

[m] ; R). We will discuss thissubspace in detail in Sect. 10.3. The set of multi-fans with integral weights forms alattice inside MultiFansλ which is a certain sublattice of Zn−1(�(n−1)

[m] ; Z).

3 Definitions: Multi-Polytopes

3.1 Multi-Polytopes

LetΔ be a simplicialmulti-fanwith characteristic function λ : [m] → V . Let HP(V ∗)denote the set of all affine hyperplanes in the dual vector space V ∗.

For each i ∈ [m] choose an affine hyperplane H(i) ⊂ V ∗ in the dual space whichis orthogonal to the linear hull of the i-th cone. In other words, H(i) is defined byequation H(i) = {u ∈ V ∗ | 〈u, λ(i)〉 = ci } for some constant ci ∈ R called thesupport parameter of H(i).

Definition 3.1 A multi-polytope P is a pair (Δ,H), where Δ is a multi-fan, andH : [m] → HP(V ∗) is a function such thatH(i) is orthogonal to λ(i) for any i ∈ [m].We say that P is based on the multi-fan Δ.

Although the definitionmay be stated in general, we restrict to simplicial multi-fansΔ, in which case P is called a simple multi-polytope.

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Let us denote the set of all multi-polytopes based on Δ by Poly(Δ). Every suchmulti-polytope is completely determined by its support parameters c1, . . . , cm . ThusPoly(Δ) has natural coordinates (c1, . . . , cm) and may be identified with R

m . Thisspace is called the space of analogous multi-polytopes based on Δ.

To simplify notation, we denote H(i) by Hi and set

HI :=⋂

i∈I Hi for I ∈ K .

HI is a codimension |I | affine subspace in V ∗, since the normals of the hyperplanesHi , i ∈ I are linearly independent by ∗-condition. In particular, when I is a maximalsimplex, I ∈ K 〈n〉, HI is a point in V ∗ which is called the vertex of P .

Definition 3.2 Let Δ be a simplicial multi-fan in V with the underlying simplicialcomplex K and let P be a simple multi-polytope based on Δ. Let I ∈ K . Consider asimple multi-polytope FI = (ΔI ,HI ) in the space HI ⊂ V ∗. Note that the projectedmulti-fan ΔI is defined in the space VI (see Sect. 2.4), so the multi-polytope basedon ΔI should formally lie in V ∗

I . Nevertheless, we may identify HI with V ∗I . The

supporting hyperplanes of FI are defined as follows:HI ( j) = HI ∩Hj for any vertexj of lkK I . The multi-polytope FI is called the face of P dual to I .

3.2 Duistermaat–Heckman Function of a Multi-Polytope

Suppose I ∈ K 〈n〉. Then the set {λ(i) | i ∈ I } is a basis of V . Denote its dual basis ofV ∗ by {uI

i | i ∈ I }, i.e. 〈uIi , λ( j)〉 = δi j where δi j denotes the Kronecker delta. Take

a generic vector v ∈ V . Then 〈uIi , v〉 = 0 for all I ∈ K 〈n〉 and i ∈ I . Set

(−1)I := (−1){i∈I |〈uIi ,v〉>0} and (uIi )

+ :={uIi if 〈uI

i , v〉 > 0,

−uIi if 〈uI

i , v〉 < 0.

We denote by C∗(I )+ the cone in V ∗ spanned by (uIi )

+’s (i ∈ I ) with apex at avertex HI of a multi-polytope P , and by φI the function on V ∗ which takes value 1inside C∗(I )+ and 0 outside (the indicator function of the cone C∗(I )+).

Definition 3.3 A function DHP on V ∗\⋃mi=1 Hi defined by

∑I∈K 〈n〉

(−1)Iw(I )φI

is called a Duistermaat–Heckman function associated with P .

The summands in the definition depend on the choice of a generic vector v ∈ V .Nevertheless, the function itself is independent of v when Δ is complete (we refer toHattori andMasuda (2003) whenw is integral-valued and note that the same argumentworks for real weights).

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The function DHP for a simplemulti-polytope P based on a complete multi-fan hasthe following geometrical interpretation. Let S be the realization of first barycentricsubdivision of K and let GI ⊂ S be the dual face of I ∈ K , I = ∅, i.e. a realizationof the set {{I < I1 < · · · < Ik} ∈ K ′}. If I ∈ K 〈n〉, then GI is a point. For a givenmulti-polytope P based on Δ there exists a continuous map ψ : S → V ∗ such thatψ(GI ) ⊂ HI for any I ∈ K , I = ∅ (in particular, when I ∈ K 〈n〉, this map sendsthe point GI ∈ S to the vertex HI of a multi-polytope P). This map is unique up tohomotopy preserving the stratifications.

Let us take any point u ∈ V ∗\⋃mi=1 Hi . Then u is not contained in the image of ψ

by the construction of ψ . Thus we may consider the induced map in homology:

ψ∗ : Hn−1(S; R) → Hn−1(V∗\{u}; R).

The underlying simplicial cycle [Δ] may be considered as an element of the groupHn−1(S; R). Since V ∗ is oriented, we have the fundamental class [V ∗\{u}] ∈Hn−1(V ∗\{u}; R). Thus

ψ∗([Δ]) = WNP (u) · [V ∗\{u}],

for some number WNP (u) ∈ R. This number has a natural meaning of windingnumber of cycle [Δ] around u. It happens that this number is exactly the value ofDHP at the point u ∈ V ∗ (see details in Hattori and Masuda 2003, Sec. 6).

It is easily seen from the above consideration that DHP has a compact support whenΔ is complete. Thus in the case of complete multi-fan we may define the volume of amulti-polytope P as

Vol(P) =∫V ∗

DHP (u)dμ (3.1)

with respect to some Euclidean measure on V ∗ (in a presence of a lattice N ⊂ V themeasure is normalized so that the fundamental domain of N∗ ⊂ V ∗ has volume 1).

Finally, we may consider the volume as a function on the space Poly(Δ) ∼= Rm

of analogous multi-polytopes. We have a function VΔ : Rm → R whose value at

(c1, . . . , cm) equals Vol(P) for the multi-polytope P with the support parametersc1, . . . , cm . The goal of the next section is to study this function using equivariantlocalization ideas and prove Theorem 1.1.

Remark 3.4 Needless to say that in case of actual simple convex polytopes the notionsintroduced above coincide with the classical ones. If P is a simple convex polytopeand Δ is its normal fan, then DHP takes the value 1 inside P and 0 outside. Thevolume of P is just the usual volume. Note that even if Δ is an actual fan, not allmulti-polytopes based on Δ are actual convex polytopes. Nevertheless, the notion ofvolume and Duistermaat–Heckman function have transparent geometrical meaningsfor all of them.

Example 3.5 Consider the two-dimensional multi-fan Δ with m = 5 and V = R2

depicted on Fig. 1, left. Its characteristic function is the following: λ(1) = (1, 0),

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Fig. 1 Example of a multi-fan Δ and a multi-polytope P based on it

Fig. 2 Duistermaat–Heckmanfunction of the multi-polytope P

H1H2

H3

H4

H5

21

1

1

11

λ(2) = (−2, 1), λ(3) = (1,−2), λ(4) = (0, 1), λ(5) = (−1,−1). The weightfunction takes the value 1 on the subsets {1, 2}, {2, 3}, {3, 4}, {4, 5}, {1, 5} and thevalue 0 on all other subsets. Geometrically this indicates the fact that in the multi-fanwe have the cones generated by {λ(1), λ(2)}, {λ(2), λ(3)}, etc. with multiplicity one,and do not have the cones generated by {λ(1), λ(3)}, {λ(1), λ(4)}, and so on. It can beseen that every generic point of V = R

2 is covered by exactly two cones, thereforeΔ is pre-complete of degree 2. Moreover, a simple check shows that all its projectedmulti-fans are complete. Hence Δ is complete. The underlying chain of Δ has theform (1, 2) + (2, 3) + (3, 4) + (4, 5) + (5, 1) ∈ C1(�(1)

[5] ; R) which is obviously asimplicial cycle. The underlying complex K of Δ is a circle made of 5 segments, and[Δ] is its fundamental class.

An example of a multi-polytope P based on Δ is shown at Fig. 1, right. Eachhyperplane Hi is orthogonal to the linear span of the corresponding ray λ(i) of Δ,i ∈ [5]. The Duistermaat–Heckman function of P is shown on Fig. 2. The function

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is constant on the chambers: it takes value 2 in the middle pentagon since the multi-polytope “winds” around the points of this region twice, and takes value 1 on trianglesadjacent to the central pentagon. The value of DHP in all other chambers is 0. Thevolume of a multi-polytope is therefore not just the volume of the five-point star: thepoints in the central region contribute to the volume twice.

4 Volume Polynomial from the Index Map

4.1 Index Map

Let Δ = (wch, λ) be a simplicial multi-fan in V ∼= Rn withm rays. The characteristic

function λ : [m] → V may be considered as a linear map λ : Rm → V which sends

the basis vector ei ∈ Rm , i ∈ [m] to λ(i). Let {xi }i∈[m] be the basis of (Rm)∗ dual

to {ei }i∈[m], so that (Rm)∗ = 〈x1, . . . , xm〉. Let us also consider the adjoint mapλ� : V ∗ → (Rm)∗. By definition it sends the vector u ∈ V ∗ to

m∑i=1

〈u, λ(i)〉xi .

For any maximal simplex I = {i1, . . . , in} ∈ K 〈n〉 the vectors {λ(i)}i∈I form a basisof V according to ∗-condition, defined in Sect. 2.2. Let {uI

i }i∈I be the dual basis ofV ∗. Let ιI : (Rm)∗ → V ∗ be the linear map defined by

ιI (xi ) ={uIi if i ∈ I,

0, if i /∈ K .(4.1)

Consider R[x1, . . . , xm], the algebra of polynomials on (Rm)∗. Also let R[V ∗] denotethe algebra of polynomials on V ∗. Both polynomial algebras are graded, where we setthe degrees of the generating spaces (Rm)∗ and V ∗ to 2. The linear map ιI inducesthe graded algebra homomorphism

ιI : R[x1, . . . , xm] → R[V ∗],

denoted by the same letter. In the following, if A is a graded algebra, we denote byA j its homogeneous part of degree j .

Let S−1R[V ∗] denote the ring of rational functions over R[V ∗] graded in a nat-

ural way. Given a weight function w : K 〈n〉 → R we can define the linear mapπΔ

! : R[x1, . . . , xm] → S−1R[V ∗] as the following weighted sum:

πΔ! (x) =

∑I∈K 〈n〉

w(I )ιI (x)

| det λI |∏i∈I ιI (xi )(4.2)

for x ∈ R[x1, . . . , xm]. We assume that an inner product is fixed on V , so that| det λI | = | det(λ(i)i∈I )| is well-defined even if there is no lattice in V . The inner

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product on V induces a Euclideanmeasure on V ∗ and | det λI | is the volume of the par-allelepiped spanned by {λ(i)}i∈I . The translation invariant measure on V ∗ is assumedthe same as in (3.1). The map πΔ

! is well-defined since λI are isomorphisms. It can beseen that πΔ

! is homogeneous of degree −2n. It is called the index map of a multi-fanΔ = (K , w, λ).

Theorem 4.1 The following properties of Δ are equivalent:1. The image of πΔ

! lies in R[V ∗] ⊂ S−1R[V ∗];

2. The underlying chain wch = ∑I∈K 〈n〉 w(I )I is closed;

3. The multi-fan Δ = (wch, λ) is complete.

Proof Equivalence of (2) and (3) was already shown in Proposition 2.7. The impli-cation (2) ⇒ (1), in case when λ takes values in the lattice and w is integer-valued,is proved in Hattori and Masuda (2003, Lm. 8.4). It should be noted that in thiscase | det λI | appearing in the denominator is nothing but the order of the finite groupGI = N/NI , where N ⊂ V is the lattice and NI is a sublattice generated by {λ(i)}i∈I .The situation when λ and w are rational is reduced to the integral case by multiplyingall values of λ and w by a common denominator (both conditions (1) and (2) areinvariant under rescaling). The real case follows by continuity. Indeed, the subset ofsimplicial cycles with rational coefficients, Zn−1(K ; Q), is dense in Zn−1(K ; R); theright hand side of (4.2) is continuous with respect to λ and w; and the subset R[V ∗]is closed in S−1

R[V ∗]. Therefore, arbitrary complete multi-fan (w, λ) can be approx-imated by a sequence of rational complete multi-fans Δα = (wα, λα) which impliesthat the values of πΔ

! are approximated by the values of πΔα

! . Since the values of πΔα

!are polynomials, so are the values of πΔ

! .Let us prove that (1) implies (2). Take any simplex J ∈ K such that |J | = n − 1

and consider the monomial xJ = ∏i∈J x j of degree 2(n − 1) lying in R[x1, . . . , xm].

The map πΔ! lowers the degree by 2n thus we have degπΔ

! (xJ ) = −2. Condition (1)implies that πΔ

! (xJ ) is a polynomial, thus πΔ! (xJ ) = 0. By definition, we have

πΔ! (xJ ) =

∑I∈K 〈n〉

w(I )ιI (xJ )

| det λI |∏i∈I ιI (xi )

Note that ιI is a ring homomorphism and ιI (xi ) = 0 if i /∈ I by (4.1). Therefore,

πΔ! (xJ ) =

∑I∈K 〈n〉,J⊂I

w(I )∏

i∈J ιI (xi )

| det λI |∏i∈I ιI (xi )=

∑j∈[m]\J,I :=J∪{ j}∈K 〈n〉

w(I )

| det λI |ιI (x j )

Recall that ιI (xi ) = uIi , where {uI

i }i∈I is the basis of V ∗ dual to the basis {λ(i)}i∈I ofV . Consider the linear functional � ∈ V ∗ taking the value �(v) = det((λ(i))i∈J , v)

for any v ∈ V . It can be seen that | det λI |ιI (x j ) ∈ V ∗, where I = J ∪ { j}, coincideswith � up to sign. More precisely | det λI |ιI (x j ) = [I : J ]�, where [I : J ] is the inci-dence sign of two simplices of K (it appears because we need to permute the vectors((λ(i))i∈J , λ( j)) in order to get the positive determinant). Therefore,

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0 = πΔ! (xJ ) = 1

�

∑I∈K 〈n〉,J⊂I

[I : J ]w(I )

It remains to notice that the sum in this expression is exactly the coefficient of J inthe simplicial chain dwch ∈ Cn−2(K ; R). This calculation applies to any J ∈ K ,|J | = n − 2, therefore dwch = 0. ��

The map λ� : V ∗ → (Rm)∗, the adjoint of λ, induces the ring homomorphismR[V ∗] → R[x1, . . . , xm]. Hence R[x1, . . . , xm] obtains the structure of R[V ∗]-module. It can be checked that λ� is the right inverse of each ιI : (Rm)∗ → V ∗,therefore all ring homomorphisms ιI : R[x1, . . . , xm] → R[V ∗] are the R[V ∗]-module homomorphisms. Thus πΔ

! is also a homomorphism of R[V ∗]-modules (evenin the case wch is not closed).

Remark 4.2 Note that conditions (1) and (2) in Theorem 4.1 make sense over anarbitrary field k. We may start with a k-valued chain wch ∈ Cn−1(K ; k) and a charac-teristic function valued in k

n . These data allow to define themaps ιI andπΔ! absolutely

similar to the real case.

Problem 4.3 Does equivalence of (1) and (2) in Theorem 4.1 hold for arbitrary fields?

For general fields we cannot reduce the task to the integral case but it is likely thatthere exists a straightforward algebraical proof.

4.2 Stanley–Reisner Rings

Let us recall the definition of the Stanley–Reisner ring.

Definition 4.4 Let K be a simplicial complex on the vertex set [m] and k be a groundring (either Z or a field). The Stanley–Reisner ring is the quotient of a polynomial ringby the Stanley–Reisner ideal:

k[K ] := k[x1, . . . , xm]/ISR, where ISR = (xi1 · . . . · xik | {i1, . . . , ik} /∈ K ),

endowed with the grading deg xi = 2 and the natural structure of gradedk[x1, . . . , xm]-module.

For now let us concentrate on the case k = R. Given a characteristic function λ

on K we may define a certain ideal in R[K ] generated by linear forms. As before, letλ� : V ∗ → (Rm)∗ = 〈x1, . . . , xm〉 denote the adjoint map of λ : R

m → V . Let Θ

denote the ideal of R[x1, . . . , xm] generated by the image of λ�. By abuse of notationwe denote the corresponding ideal in R[K ] with the same letter Θ .

Let us state things in the coordinate form. Fix a basis f1, . . . , fn of V . Then everycharacteristic value λ(i), i ∈ [m] is written as a row-vector (λi,1, . . . , λi,n), whereλi, j ∈ R. The ∗-condition for λ (see Sect. 2.2) states that the square matrix formed byrow-vectors (λi,1, . . . , λi,n)i∈I is non-degenerate for any I ∈ K 〈n〉.

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If we consider the dual basis f1, . . . , fn in the dual space V ∗, then its image underλ� : V ∗ → (Rm)∗ = 〈x1, . . . , xm〉 has the form

θ j := λ�( f j ) = λ1, j x1 + λ2, j x2 + · · · + λm, j xm

for j = 1, . . . , n. Thus Θ (as an ideal either in R[x1, . . . , xm] or R[K ]) is generatedby the elements θ1, . . . , θn . In particular, if λ is integer-valued, then Θ = (θ1, . . . , θn)

may be considered as a well-defined ideal in Z[K ] or Z[x1, . . . , xm].It is known that the Krull dimension of R[K ] equals dim K + 1 = n (see e.g.

Stanley 1996), and θ1, . . . , θn is a linear system of parameters in R[K ] for any charac-teristic function λ and every choice of a basis in V (e.g. Buchstaber and Panov 2015,Lm. 3.3.2). Thus R[K ]/Θ has Krull dimension 0, which in our case is equivalent tosaying that R[K ]/Θ is a finite-dimensional vector space. Moreover, it is known (seee.g. Hattori and Masuda 2003, Lm. 8.1 or Ayzenberg 2016, Lm. 3.5) that the classesof monomials xI = xi1 · . . . · xik taken for each simplex I = {i1, . . . , ik} ∈ K linearlyspan R[K ]/Θ (however there exist relations on these classes!).

We introduce the following notation to make the exposition consistent with that ofHattori and Masuda (2003):

H∗T (Δ; k) := k[K ], H∗(Δ; k) := k[K ]/Θ, (4.3)

and, for short, H∗T (Δ) := H∗

T (Δ; R) and H∗(Δ) := H∗(Δ; R).

4.3 Evaluation on Fundamental Class

Let x = x j1i1

· · · · · x jkik

be a monomial whose index set {i1, . . . , ik} is not a simplex

of K . Then ιI (x) = 0 for any I ∈ K 〈n〉, according to (4.1). Therefore πΔ! (x) = 0.

Hence πΔ! vanishes on the Stanley–Reisner ideal ISR and descends to the map

πΔ! : R[K ] → R[V ∗].

Since πΔ! is a map of R[V ∗]-modules, we may apply ⊗R[V ∗]R to πΔ

! . This gives alinear map

∫Δ

: H2n(Δ) → R ∼= R[V ∗]/R[V ∗]+

Definition 4.5 Let Δ = (K , w, λ) be a complete simplicial multi-fan. The map∫Δ

: H2n(Δ) → R is called “the evaluation on the fundamental class of Δ”.

We denote the composite map R[x1, . . . , xm] � H2n(Δ) → R by∫Δ,R[m].

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4.4 Chern Class of a Multi-Polytope

Let P be a multi-polytope based on a complete simplicial multi-fan Δ = (K , w, λ)

of dimension n, and let c1, . . . , cm ∈ R be the support parameters of P . The element

c1(P) := c1x1 + · · · + cmxm ∈ H2(Δ)

is called the first Chern class of P .

Proposition 4.6

Vol(P) = 1

n!∫

Δ

c1(P)n . (4.4)

Proof If λ and w are integral, the statement is proved in Hattori and Masuda (2003,Lm. 8.6). The rational case follows from the integral case by the following arguments.(1) In the rational case we may choose a refined lattice such that λ becomes integralwith respect to this lattice (this would change the Euclidean measure on V ∗, but thischange affects both sides of (4.4) in the same way). (2) A rational weight w may beturned into an integral weight by rescaling (both sides of (4.4) depend linearly on w,thus rescaling of w preserves (4.4)). Real case follows by continuity, since both sidesof (4.4) depend continuously on λ and w. ��

It is easily seen that, for a given Δ, the expression on the right hand side of (4.4) isa homogeneous polynomial of degree n in the variables c1, . . . , cm :

VΔ(c1, . . . , cm) = 1

n!∫

Δ

(c1x1 + · · · + cmxm)n .

Thus Proposition 4.6 implies Theorem 1.1.

5 Basic Properties of Volume Polynomials

5.1 Partial Derivatives of Volume Polynomial

We continue to assume that there is a fixed inner product in V which makes theintegral lattice in V unnecessary. The inner product allows to identify V and V ∗ andto introduce a measure on each affine subspace of V or V ∗. Consider the space ΛkVof exterior forms on V . Given an inner product in V we obtain an inner product onΛkV .

Suppose that every simplex I ∈ K is oriented somehow. For a characteristic func-tion λ : [m] → V on K and I = {i1, . . . , ik} ∈ K let λ(I ) denote the skew formλ(i1) ∧ · · · ∧ λ(ik) ∈ ΛkV , where (i1, . . . , ik) is the positive order of vertices of I(that is the order which defines the positive orientation of a simplex). Denote the normof λ(I ) by covol(I ):

covol(I ) := ‖λ(I )‖ = ‖λ(i1) ∧ · · · ∧ λ(ik)‖.

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Recall from Sect. 3 the notion of a face of a multi-polytope. If P is a multi-polytopeof dimension n and I ∈ K then FI is a multi-polytope of dimension n − |I | sittingin the affine subspace HI ⊂ V ∗. There is a measure on HI determined by the innerproduct, hence we may define the volume of FI . The following lemma shows that wecan compute the volumes of faces from the volume polynomial.

Lemma 5.1 (cf. Timorin 1999, Thm. 2.4.3) Let J ⊂ [m] and let ∂J denote the dif-ferential operator

∏i∈J ∂i . Consider the homogeneous polynomial ∂J VΔ of degree

n − |J |. Then1. Let θu denote the linear differential operator

∑mi=1〈u, λ(i)〉∂i for u ∈ V ∗. Then

θuVΔ = 0.2. If J /∈ K , then ∂J VΔ = 0;3. If J ∈ K , then the value of the polynomial ∂J VΔ at a point (c1, . . . , cm) ∈ R

m isequal to

Vol FJ

covol(J )(5.1)

when |J | < n and

w(J )

covol(J )= w(J )

| det λJ | (5.2)

when |J | = n. Here ci are the support parameters of a multi-polytope P and FJ

are its faces.

Proof (1) We have

θuVΔ = 1

n!∫ (

m∑i=1

〈u, λ(i)〉 ∂

∂ci

)(c1x1 + · · · + cmxm)n

= 1

(n − 1)!∫ (

m∑i=1

〈u, λ(i)〉xi)

· (c1x1 + · · · + cmxm)n−1 = 0,

since∑m

i=1〈u, λ(i)〉xi = 0 in H∗(Δ).(2) The proof of second statement is completely similar to (1). We have

∂J VΔ = 1

n!∫ (∏

i∈J

∂

∂ci

)(c1x1 + · · · + cmxm)n

= 1

(n − |J |)!∫ (∏

i∈J

xi

)· (c1x1 + · · · + cmxm)n−|J | = 0,

since xJ = ∏i∈J xi = 0 in H∗(Δ) for J /∈ K .

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(3) The second claim requires some technical work. At first, let |J | = n, i.e.J ∈ K 〈n〉. We have

∂J VΔ = ∂J1

n!∫

Δ

(c1x1 + · · · + cmxm)n =∫

Δ

xJ = πΔ! (xJ ),

where xJ = ∏i∈J xi . By the definition of the index map (4.2) we have

πΔ! (xJ ) =

∑I∈K 〈n〉

w(I )ιI (xJ )

| det λI |∏i∈I ιI (xi ).

If I = J , the corresponding summand vanishes, since ιI (x j ) = 0 for j /∈ I by (4.1).The summand corresponding to I = J contributes w(J )

| det λJ | which proves the statement.Let us prove the case |J | < n. Recall that the projected multi-fan ΔJ =

(lkK J, wJ , λJ ) is the multi-fan in the vector space VJ = V/〈λ( j) | j ∈ J 〉. Thereexists a “restriction” map

ϕJ : H∗(Δ) → H∗(ΔJ ),

defined as follows:

ϕJ (x j ) =

⎧⎪⎨⎪⎩x j , if j ∈ lkK J ;−∑

i∈lkK J pJi, j xi , if j ∈ J ;

0, otherwise

Here the constants pJi, j for j ∈ J and i ∈ lkK J are defined by

projJ λ(i) =∑j∈J

pJi, jλ( j), (5.3)

where projJ λ(i) is the orthogonal projection of the vector λ(i) to the linear subspacespanned by λ( j) ( j ∈ J ).

The homomorphism ϕJ is now defined on the level of polynomial algebras.

Claim 5.2 ϕJ is a well-defined ring homomorphism from H∗(Δ) = R[K ]/Θ toH∗(ΔJ ) = R[lkK J ]/ΘJ .

Proof The proof is a routine check. First let us prove that Stanley–Reisner relationsin Δ are mapped to the Stanley–Reisner ideal of ΔJ . Let I be a non-simplex ofK . The definition of ϕJ implies that ϕJ (xI ) = 0 unless I ⊂ J ∪ Vert(lkK J ). IfI ⊂ J∪Vert(lkK J ), we have that I∩Vert(lkK J ) is a non-simplex of lkK J (otherwisewe would have I ∈ K contradicting the assumption). Then the element ϕJ (xI ) =ϕJ

(∏i∈I∩Vert(lkK J ) xi

) · ϕJ(∏

i∈I∩J xi) = ∏

i∈I∩Vert(lkK J ) xi · ϕJ(∏

i∈I∩J xi)lies

in the Stanley–Reisner ideal of lkK J .

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Let us check that linear relations in H∗(Δ) are mapped into linear relations ofH∗(ΔJ ). A general linear relation in H∗(Δ) has the form

∑i∈[m]〈u, λ(i)〉xi for some

u ∈ V ∗. The map ϕJ sends it to the element

∑i∈Vert(lkK J )

⎛⎝〈u, λ(i)〉 −

∑j∈J

pJi, j 〈u, λ( j)〉

⎞⎠ xi

=∑

i∈Vert(lkK J )

⟨u, λ(i) −

∑j∈J

pJi, jλ( j)

⟩xi =

∑i∈Vert(lkK J )

〈u, λJ (i)〉xi .

(note that λ(i)−∑j∈J pJ

i, jλ( j) = λ(i)−projJ λ(i) = λJ (i) is the projection of λ(i)to the plane orthogonal to 〈λ( j)〉 j∈J ). The last expression is zero in H∗(ΔJ ). ��

Next we show that restriction homomorphism is compatible with the first Chernclasses of the multi-polytopes.

Claim 5.3 ϕJ sends c1(P) to c1(FJ ).

Proof Recall that HJ denotes the ambient space of the face FJ of the multi-polytopeP . The supporting hyperplanes of FJ are given by intersections HJ ∩ Hi , where Hi

is the supporting hyperplane of P for i ∈ lkK J .Let us denote by UJ the subspace spanned by λ( j)’s ( j ∈ J ) so that VJ = V/UJ .

By the definition (see Sect. 2.4), λJ (i) is the projection image of λ(i) on VJ if i isthe vertex of lkK J . As in the proof of previous claim we identify the quotient spaceVJ = V/UJ with the orthogonal complement U⊥

J of UJ . The projected vector λJ (i)can be considered as the element in V and we have

λ(i) = λJ (i) + projJ λ(i) (5.4)

with respect to the orthogonal decomposition V = U⊥J ⊕UJ .

The affine hyperplane Hi is given by {u ∈ V ∗ | 〈u, λ(i)〉 = ci }. The affine planeHJ is given by {u ∈ V ∗ | 〈u, λ( j)〉 = c j , for all j ∈ J }. By using (5.3) and (5.4) wemay write the intersection Hi ∩ HJ as

{u ∈ HJ | 〈u, λJ (i) + projJ λ(i)〉 = ci }

=⎧⎨⎩u ∈ HJ

∣∣∣∣⟨u, λJ (i) +

∑j∈J

pJi, jλ( j)

⟩= ci

⎫⎬⎭

=⎧⎨⎩u ∈ HJ | 〈u, λJ (i)〉 = ci −

∑j∈J

pJi, j c j

⎫⎬⎭ .

Therefore the i-th support parameter of FJ is ci −∑j∈J pJ

i, j c j for i ∈ lkK J . Nowit remains to note that the coefficient of xi in the projected class ϕJ (c1(P)) is exactlyci − ∑

j∈J pJi, j c j . Thus ϕJ (c1(P)) = c1(FJ ). ��

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Now we prove the following

Claim 5.4

∫Δ

y∏j∈J

x j = 1

covol(J )

∫ΔJ

ϕJ (y) for any y ∈ H∗(Δ).

Proof Let us denote by Vol S the volume of the parallelepiped formed by a set ofvectors S. Then covol(J ) = Vol{λ(i)}i∈J and the index map can be written as

πΔ! (x) =

∑I∈K 〈n〉

w(I )ιI (x)

Vol{λ(i)}i∈I ∏i∈I ιI (xi ). (5.5)

Let I ∈ lkK J and, therefore, I � J ∈ K . Then

Vol{λ(i)}i∈ I�J = Vol{λ(i)}i∈J · Vol{λJ (i)}i∈ I = covol(J ) · Vol{λJ (i)}i∈ I .

This together with (5.5) implies the lemma. ��Applying Claim 5.4 to y = c1(P)n−|J | and using Claim 5.3, we obtain

∂J VΔ = 1

(n − |J |)!∫

Δ

c1(P)n−|J | ∏j∈J

x j = 1

covol(J )

1

(n − |J |)!∫

ΔJ

c1(FJ )n−|J |.

Expression at the right evaluates to Vol(FJ )covol(J )

which finishes the proof of Lemma 5.1.

Corollary 5.5 Let Δ = (wch, λ) be a complete multi-fan. Then VΔ = 0 implieswch = 0.

Proof If VΔ = 0, then ∂J VΔ = 0 for any J ∈ K 〈n〉. This implies wch = 0. ��Remark 5.6 Of course, according to Proposition 7.2 the polynomial VΔ is non-zeroif and only if the map

∫Δis non-zero. The fact that

∫Δis non-zero for every non-

zero wch is proved by applying this map to all monomials xI , I ∈ K 〈n〉 (recall thatthese monomials span H2n(Δ)). This procedure is essentially the same as applyingdifferential operators ∂I to VΔ.

Corollary 5.7 Let ∂P denote the linear differential operator∑

i∈[m] ci∂i where ci arethe support parameters of a multi-polytope P. Then

1

n!∂nPVΔ = Vol(P),

1

(n − |I |)!∂n−|I |P ∂I VΔ = Vol(FI )

covol(I ).

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Proof Both formulas follow from Lemma 5.1 and a simple observation: if Ψ ∈R[c1, . . . , cm]k is a homogeneous polynomial of degree k, then

⎛⎝∑

i∈[m]ci∂i

⎞⎠

k

Ψ = k!Ψ (c1, . . . , cm).

(evaluation at a point coincides with the result of differentiation up to k!). ��

5.2 Recovering Multi-Fans from Volume Polynomials

When we associate a volume polynomial to a complete simplicial multi-fan, the num-bering of the one-dimensional cones by [m] is incorporated in the data of themulti-fan.We call a multi-fan with the numbering a based multi-fan. Two based multi-fans Δ

and Δ′ are said to be equivalent if there is an automorphism of V which induces anisomorphism between Δ and Δ′ preserving the numbering. In the presence of a latticeN ⊂ V there should be an automorphism of the lattice with this property. Equivalentcomplete simplicial based multi-fans have the same volume polynomial. We will seethat the converse holds for complete simplicial based multi-fans Δ whose underlyingsimplicial complexes are oriented strongly connected pseudo-manifolds. Strong con-nectedness of K means that for any two maximal simplices I, I ′ ∈ K 〈n〉 there exists asequence of maximal simplices I = I0, I1, . . . , Ik = I ′ such that |Is ∩ Is+1| = n − 1for 0 � s � k − 1.

We assume that the volume polynomial VΔ associated to Δ is non-zero. Then theclass [Δ] is non-zero. Since K is assumed to be a pseudo-manifold, w(I ) = 0 for anyI ∈ K 〈n〉. Then Lemma 5.1 shows that VΔ recovers K .

Remember that

m∑i=1

〈u, λ(i)〉xi = 0 in H∗(Δ) for any u ∈ V ∗. (5.6)

Let J ∈ K , |J | = n − 1. Since K is assumed to be a pseudo-manifold, there areexactly two elements i1 and i2 in [m] such that J ∪ {i1} and J ∪ {i2} are in K 〈n〉.Multiplying xJ = ∏

i∈J xi to the both sides in (5.6), we obtain

∑j∈J

〈u, λ( j)〉x j xJ + 〈u, λ(i1)〉xi1xJ + 〈u, λ(i2)〉xi2xJ = 0 for all u ∈ V ∗.

Applying∫Δto the above identity, we have

⟨u,∑j∈J

(∫Δ

x j xJ

)λ( j) +

(∫Δ

xi1xJ

)λ(i1) +

(∫Δ

xi2xJ

)λ(i2)

⟩= 0.

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Since this holds for all u ∈ V ∗, one can conclude

∑j∈J

(∫Δ

x j xJ

)λ( j) +

(∫Δ

xi1xJ

)λ(i1) +

(∫Δ

xi2xJ

)λ(i2) = 0. (5.7)

Note that the numbers∫Δxi1xJ and

∫Δxi2xJ are non-zero. Identity (5.7) shows that

once basis vectors {λ(i)}i∈I for some I ∈ K 〈n〉 are determined, then the other vectorsλ(k)’s will be determined by the intersection numbers

∫ΔxI where I consists of

elements in [m] with |I| = n (an element in I may appear more than once). On theother hand, since

VΔ = 1

n!∫

Δ

(c1x1 + · · · + cmxm)n,

the coefficient of cI agrees with∫ΔxI up to some non-zero constant independent of

Δ. These show that VΔ determines Δ up to equivalence.

Proposition 5.8 Two complete simplicial toric varieties are isomorphic if and only iftheir volume polynomials agree up to permutations of variables. Here it is assumedthat all λ(i)’s are the primitive generators of the rays.

Proof This follows from the above observation and the fact that two toric varieties areisomorphic if and only if their fans are isomorphic (Berchtold 2003).1 ��

6 A Formula for the Volume Polynomial

We say that the set S of n + 1 vectors in V ∼= Rn is in general position, if any n of

them are linearly independent. Any such set determines a multi-fan whose underlyingsimplicial complex is a boundary of a simplex K = ∂�[n+1]. The weights of allmaximal simplices are the same up to sign due to closedness condition dwch =0. Thus without loss of generality we may assume that every weight is 1 or −1depending on the orientation. We call such multi-fan an elementary multi-fan anddenote it Δel(S).

Lemma 6.1 Let Δ be an elementary multi-fan determined by the vectors λ(1), . . . ,λ(n + 1) ∈ V . Let 0 = (α1, . . . , αn+1) ∈ R

n+1 be a nonzero linear relation on thesevectors, i.e.

∑n+1i=1 αiλ(i) = 0. Then

VΔ(c1, . . . , cn+1) = const ·(α1c1 + · · · + αn+1cn+1)n . (6.1)

for some constant const.

We postpone the proof to Sect. 8.3.

1 We are grateful to Ivan Arzhantsev from whom we learned this fact.

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Remark 6.2 It is not difficult to compute the constant: just apply the differential oper-ator ∂J for J ⊂ [n + 1], |J | = n to both sides of (6.1) and use Lemma 5.1. However,we do not need this constant at the moment and ignore it to simplify the exposition.

Theorem 6.3 Let Δ = (wch, λ) be a complete multi-fan. Let v ∈ V be a genericvector. Then

VΔ(c1, . . . , cm) = 1

n!∑

I={i1,...,in}∈K

w(I )

| det λI |∏nj=1 αI, j

(αI,1ci1 + · · · + αI,ncin )n,

(6.2)

where αI,1, . . . , αI,n are the coordinates of v in the basis (λ(i1), . . . , λ(in)), andw(I )is the weight.

Proof We derive a more general family of formulas, and (6.2) will be a particular case.Let [m′] be a set containing [m] and let

zch =∑

J∈([m′]n+1)

z(J )J ∈ Cn(�[m′]; R)

be a simplicial chain such that dzch = wch (it exists since wch , considered as anelement in Cn(�[m′]; R), is closed hence exact). Consider any function η : [m′] → Vwhich extends λ : [m] → V and satisfies the condition: for any J = { j1, . . . , jn+1}with z(J ) = 0 the vectors η( j1), . . . , η( jn+1) are in general position. Thus for anysuch J we can construct an elementary multi-fan Δel(η(J )).

In the group of multi-fans we have a relation Δ = ∑J∈([m′]

n+1)z(J )Δel(η(J )), if Δ

is considered as a multi-fan on [m′] (recall that multi-fans with the same characteristicfunction can be added to each other and multiplied by integers by performing thisoperations on their weights, and therefore such multi-fans form an abelian group).Volume polynomial is additive, hence we get

VΔ =∑

J∈([m′]n+1)

z(J )VΔel (η(J )). (6.3)

Therefore, any simplicial chain whose boundary is wch gives a formula for thevolume polynomial. Now let us consider the particular case, namely, the cone overwch . Let [m′] = [m] � {r} and set η(r) = v, for a generic vector v ∈ V . So the phrase“v is a generic vector” means that the set λ(I ) � {v} is in general position for anyI ⊂ [m] such that |I | = n and w(I ) = 0. The function z on the cone is defined in anobvious way: z(I � {r}) := w(I ).

Relation (6.3) and Lemma 6.1 imply

VΔ =∑

I={i1,...,in}⊂[m]const ·w(I ) · (αI,1ci1 + · · · + αI,ncin + βI cr )

n . (6.4)

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The tuple (αI,1, . . . , αI,n, βI ) is a linear relation on the vectors λ(i1), . . . , λ(in), v.Therefore we may assume that βI = −1 and (αI,1, . . . , αI,n) are the coordinates of v

in the basis λ(i1), . . . , λ(in).Left hand side of (6.4) does not depend on cr (it is a redundant support parameter),

therefore we may put cr = 0:

VΔ =∑

I={i1,...,in}⊂[m]AI · w(I ) · (αI,1ci1 + · · · + αI,ncin )

n . (6.5)

To compute the constants AI take any J = { j1, . . . , jn} ∈ K and apply the dif-ferential operator ∂J = ∂

∂c j1· · · · · ∂

∂c jnto the identity (6.5). On the left we have

∂J VΔ = w(J )| det λJ | , according to Lemma 5.1. On the right side all summands with

I = J vanish, and the one with I = J contributes n! · AJ · w(J )∏

i αJ,i . ThusAJ = 1

n!1

| det λJ |·∏i αJ,iand the statement follows. ��

Remark 6.4 Note that the formula (6.2) can be applied to compute the volume of asimple convex polytope in the casewhen the polytope is described as the intersection ofhalf-spaces with the given equations. In this case the formula is known as Lawrence’sformula (Lawrence 1991). It has found applications in explicit volumes’ calculations.

Example 6.5 Consider the standard fan Δ of CP2, generated by the vectors λ(1) =(1, 0), λ(2) = (0, 1), λ(3) = (−1,−1). Take the generic vector v = (1, 2). We have

v = λ(1) + 2λ(2) = λ(2) − λ(3) = −2λ(3) − λ(1).

Theorem 6.3 implies

VΔ = 1

2

(1

2(c1 + 2c2)

2 − (c2 − c3)2 + 1

2(−c1 − 2c3)

2)

.

This expression equals 12 (c1 +c2 +c3)2. The same expression is given by Lemma 6.1.

Example 6.6 Consider the normal fan of the standard n-cube. The underlying simpli-cial complex is isomorphic to the boundary of cross-polytope. Let {1, . . . , n,−1, . . . ,−n} be its set of vertices, so the maximal simplices have the form {±1, . . . ,±n}. Wehave λ(±i) = ±ei . Take the generic vector v = e1 + · · · + en . Then Theorem 6.3implies

VΔ = 1

n!∑

(ε1,...,εn)∈{+,−}n1∏n

i=1 εi(ε1cε11 + · · · + εncεnn)

n .

On the other hand, we have VΔ = ∏i (ci + c−i ) by geometrical reasons. Indeed, the

polytope dual to Δ is the brick with sides {ci + c−i }i∈[n]. By setting c−i = 0 for eachi we get the identity

123


n∏i=1

ci = 1

n!∑I⊆[n]

(−1)n−|I |cnI , (6.6)

where cI = ∑i∈I ci . This identity is well known as the discrete polarization identity.

Remark 6.7 The proof of Theorem 6.3 implies the following consideration. Take twosimplicialn-chains zch,1, zch,2 ∈ Cn(�[m]; R) endowedwith functionsη1, η2 : [m] →Rn such that ηε(J ) is in general position for any simplex J of the chain zch,ε , ε =

1, 2. Assume that dzch,1 = dzch,2 and the functions η1, η2 agree on the vertices ofthe boundary. Then the volume polynomial of the multi-fan Δ = (dzch,1, η1) =(dzch,2, η2) can be expressed by two formulas:

∑J=( j1,..., jn+1)⊂[m′]

const ·z1(J )(αJ,1c j1 + · · · + αJ,n+1c jn+1)n = VΔ

=∑

J=( j1,..., jn+1)⊂[m′]const ·z2(J )(αJ,1c j1 + · · · + αJ,n+1c jn+1)

n .

We may take a difference of the left and right parts and summarize as follows. Let ustake any closed simplicial n-chain zch , dzch = 0, on the vertex set [m′], and endow itwith a function η : [m′] → R

n which is in general position on any simplex J of thechain. Then we get an identity

∑J=( j1,..., jn+1)⊂[m′]

const ·z(J )(αJ,1c j1 + · · · + αJ,n+1c jn+1)n = 0

(the constants may be computed by the same method as we used previously). Thisseems to be a quite general way to construct algebraical identities from geometricaldata.

This idea can be illustrated by a simple identity obtained in Example 6.5:

1

2(c1 + 2c2 − c4)

2 − (c2 − c3 − c4)2 + 1

2(−c1 − 2c3 − c4)

2 = (c1 + c2 + c3)2.

This identity is induced by the schematic picture shown on Fig. 3. Note that the laststep in the proof of Theorem 6.3 was to specialize c4 = 0, but even without thisspecialization the identity holds true.

Fig. 3 Two simplicial chains with vector functions having the same boundary

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7 Poincare Duality Algebra of a Multi-Fan

7.1 Poincare Duality Algebras

Definition 7.1 Let k be a field. Let A∗ = ⊕nj=0 A2 j be a finite-dimensional graded

commutative k-algebra such that

– there exists an isomorphism∫A : A2n → k;

– the pairing A2p ⊗ A2n−2p → k, a ⊗ b → ∫A(a · b) is non-degenerate.

Then A is called a Poincare duality algebra of formal dimension 2n.

Let ∂i = ∂∂ci

, i ∈ [m] be the differential operator acting on the ring of polynomialsR[c1, . . . , cm] in a standard way. For a subset I ⊂ [m] let ∂I denote the product∏

i∈I ∂i .Consider the algebra of differential operators with constant coefficients D :=

R[∂1, . . . , ∂m]. It will be convenient to double the degree, so we assume deg ∂i = 2,i ∈ [m] (while still assuming that deg ci = 1). For any non-zero homogeneous poly-nomial Ψ ∈ R[c1, . . . , cm] of degree n we may consider the following ideal in D:

AnnΨ := {D ∈ D | DΨ = 0}.

It is not difficult to check that the quotient D/AnnΨ is a Poincare duality algebraof formal dimension 2n (see Timorin 1999, Prop. 2.5.1), where the “integration map”assigns the number DΨ ∈ R to any differential operator of rank n (i.e. of formaldegree 2n in our setting).

It happens that every Poincare duality algebra generated by degree two can beobtained by this construction as the following proposition shows.

Proposition 7.2 Suppose char k = 0 and let k[m] = k[x1, . . . , xm] be a polyno-mial ring, where deg xi = 2. Then the following three sets of objects are naturallyequivalent:1. Poincare duality algebras A∗ of formal dimension 2n which are the quotients of

the polynomial ring k[m];2. Non-zero homogeneous polynomials Ψ ∈ k[c1, . . . , cm] of degree n (where

deg ci = 1) up to multiplication by a non-zero constant;3. Non-zero linear maps

∫ : k[m]2n → k up to multiplication by a non-zero constant.

Proof We give a very brief sketch of the proof. For details the reader is referred to themonograph (Meyer and Smith 2005) which, among other things, describes the casechar k = 0 (for general fields instead of a polynomial Ψ one should take an elementof divided power algebra). Also we would like to mention that an equivalence of (1)and (2) is a manifestation of the well-known phenomenon called Macaulay duality (orits extended version, Matlis duality).

123


(1) ⇒ (3). Let A∗ ∼= k[m]/I be a Poincare duality quotient of the ring of polyno-mials. Then we have a linear isomorphism

∫A : A2n → k. The composite

k[m]2n � A2n → k

is the required linear map.(3) ⇒ (1). Given a linear map

∫ : k[m]2n → k we may define a pairing k[m]2p ⊗k[m]2n−2p → k by a ⊗ b → ∫

a · b. This pairing is degenerate and we define itskernel:

W ∗ =⊕p

W 2p, W 2p ={x ∈ k[m]2p |

∫x · k[m]2n−2p = 0

}.

It is easy to check that W ∗ ⊂ k[m] is an ideal and k[m]/W ∗ is a Poincare dualityalgebra.

(3) ⇒ (2). We construct a polynomial Ψ in c1, . . . , cm by

Ψ := 1

n!∫

(c1x1 + · · · + cmxm)n .

This polynomial is non-zero. Indeed, k[m]2n is additively generated by the monomialsof degree n in the variables xi . Eachmonomial can be expressed as a linear combinationof expressions of the form (c1x1+· · ·+cmxm)n for some constants ci as follows fromthe polarization identity (see (6.6) in Example 6.6). Thus expressions of the form(c1x1 + · · · + cmxm)n linearly span k[m]2n and therefore, since

∫is non-zero, the

polynomial Ψ is not a constant zero as well.(2) ⇒ (1). Given a homogeneous polynomial Ψ in the variables c1, . . . , cm , we

may construct a Poincare duality quotient k[∂1, . . . , ∂m]/AnnΨ , where the action of∂i = ∂

∂cion polynomials is defined formally in the usual way.

The consistency of all these constructions is a routine check. ��The same arguments can be used to prove that there is a one-to-one correspondence

between Poincare duality quotients of formal dimension 2n of an algebra B∗ and thenon-zero linear functionals on the linear spaceB2n . For this correspondence we do notneed the assumptions that B is generated by degree 2 and char k = 0. This motivatesthe following definition.

Definition 7.3 Let B∗ = ⊕j B2 j be a graded commutative k-algebra and suppose

that for some n > 0 a non-zero linear map∫ : B2n → k is given. The corresponding

Poincare duality quotient of B∗, i.e. the algebra

B∗/W ∗, W 2p ={b ∈ B2p |

∫b · B2n−2p = 0

}.

is denoted by PD(B∗,∫) and called Poincare dualization of B∗ (w.r.t.

∫).

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Lemma 7.4 Consider two algebras B∗1,B∗

2 with the given non-zero linear maps∫1 : B2n

1 → k,∫2 : B2n

2 → k. Let ϕ : B∗1 � B∗

2 be an epimorphism of algebrasconsistent with the integration maps: ∫2 ◦ϕ|B2n

1= ∫

1. Then ϕ induces an isomorphism

PD(B∗1,∫1)

∼= PD(B∗2,∫2).

Proof From the surjectivity of ϕ it easily follows that the kernel W ∗1 of the intersec-

tion pairing in the first algebra maps to the kernel W ∗2 of the second algebra. Thus

the homomorphism ϕ : PD(B∗1,∫1) → PD(B∗

2,∫2) is well defined. Obviously it is

surjective. Let us prove that ϕ is injective. The map ϕ is an isomorphism in degree 2n.Suppose that ϕ(a) = 0 for some 0 = a ∈ PD(B∗

1,∫1)2p. By the definition of Poincare

duality algebra, there exists b ∈ PD(B∗1,∫1)2n−2p such that ab = 0. But then we have

ϕ(ab) = ϕ(a)ϕ(b) = 0 which gives a contradiction. ��In the following let A∗(Ψ ) = D/AnnΨ denote the Poincare duality algebra cor-

responding to the homogeneous polynomial Ψ of degree n.

7.2 Algebras Associated with Multi-Fans

The linear maps∫Δ

: H2n(Δ) → R and∫Δ,R[m] : R[x1, . . . , xm]2n → R are consis-

tent with the natural projection R[x1, . . . , xm] → H∗(Δ). Thus Lemma 7.4 impliesan isomorphism

PD

(H∗(Δ),

∫Δ

)∼= PD

(R[m],

∫Δ,R[m]

).

According to the constructionsmentioned in the proof of Proposition 7.2, this Poincareduality algebra is also isomorphic toA∗(VΔ) = D/Ann VΔ, where VΔ is the volumepolynomial.

Definition 7.5 Let Δ be a complete simplicial multi-fan of dimension n with m rays.Then the algebra

A∗(Δ) := D/Ann VΔ∼= PD

(H∗(Δ),

∫Δ

)∼= PD

(R[m],

∫Δ,R[m]

)

is called a multi-fan algebra of Δ.

Remark 7.6 The constructions above show that there is a ring epimorphism fromH∗(Δ) ∼= R[K ]/Θ to A∗(Δ) ∼= PD(H∗(Δ),

∫Δ), sending xi to ∂i for each i ∈ [m].

Therefore A∗(Δ) can be considered as a quotient of H∗(Δ), and all the relations inR[K ]/Θ are inherited by A∗(Δ). We have

∂J VΔ = 0 for J /∈ K (Stanley–Reisner relations),⎛⎝∑

i∈[m]λi, j∂i

⎞⎠ VΔ = 0 for j = 1, . . . , n (Linear relations).

This proves points 1 and 2 of Lemma 5.1 in a more conceptual way.

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8 Structure of Multi-Fan Algebra in Particular Cases

8.1 Ordinary Fans

As was mentioned in the introduction, when Δ is a normal fan of a simple convexpolytope P , the construction of the algebra A∗(Δ) = D/Ann VΔ was introducedby Khovanskii and Pukhlikov (1992b) and studied extensively by Timorin (1999).In this case the underlying simplicial complex of Δ is a sphere and the weightfunction takes value +1 on all maximal simplices of K . Using purely combinato-rial and geometrical considerations Timorin proved that A∗(Δ) ∼= R[K ]/Θ . Thismeans, in particular, that dimensions di = dimA2i (Δ) are equal to hi , the h-numbersof K (see the definition below). The developed technique is applied to prove thatA∗(Δ) is a Lefschetz algebra, meaning that there exists an element ω ∈ A2(Δ) suchthat

×ωn−2k : A2k → A2n−2k

is an isomorphism for each k = 0, . . . , [n/2]. In particular this implies that the distri-bution of h-numbers of convex simplicial spheres is unimodal, i.e.

h0 � h1 � · · · � h[n/2] = hn−[n/2] � · · · � hn−1 � hn .

According toTimorin’s result, Lefschetz elementωmaybe chosen in the form c1(P) =c1∂1 + · · · + cm∂m ∈ A2(Δ) where P is any convex simple polytope with the normalfan Δ and support parameters c1, . . . , cm .

For complete non-singular fans the algebra A∗[Δ] ∼= R[K ]/Θ coincides withthe cohomology algebra H∗(XΔ; R) of the corresponding toric variety. It was theoriginal observation of Stanley (1980), that in the case when a fan Δ is poly-topal, the corresponding complete toric variety XΔ is projective, therefore thereexists a Lefschetz element in its cohomology ring according to hard Lefschetztheorem.

After Stanley’s work, several approaches were developed to prove the existence ofLefschetz elements in elementary terms, i.e. without referring to hard Lefschetz theo-rem. These approaches include in particular McMullen’s construction of the polytopealgebra (McMullen 1993), the approach based on continuous piece-wise polynomialfunctions (Brion 1996), and the approach based on the volume polynomial and differ-ential operators (Timorin 1999).

We will see that ordinary fans are not the only examples of multi-fans for which thestructure of A∗(Δ) can be explicitly described. On the other hand, A∗(Δ) is alwaysa Poincare duality algebra, so it is natural to ask if it is Lefschetz (or at least if thedimension vector (d0, d1, . . . , dn) is unimodal). We will show that this is not true ingeneral, see Theorem 10.1 below.

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8.2 Combinatorial Preliminaries

For nowwe concentrate onmulti-fans based on oriented pseudomanifolds as describedin Example 2.9. Let K be a pure simplicial complex of dimension n − 1 on the vertexset [m].

Let f j denote the number of j-dimensional simplices of K for j = −1, 0, . . . , n−1,in particular we assume that f−1 = 1 (this reflects the fact that the empty simplexformally has dimension −1). The h-numbers of K are defined by the formula:

n∑j=0

h j tn− j =

n∑j=0

f j−1(t − 1)n− j , (8.1)

where t is a formal variable. Let β j (K ) denote the reduced Betti number dim H j (K )

of K . The h′- and h′′-numbers of K are defined by the formulas

h′j = h j +

(n

j

)⎛⎝ j−1∑

s=1

(−1) j−s−1βs−1(K )

⎞⎠ for 0 � j � n; (8.2)

h′′j = h′

j −(n

j

)β j−1(K ) = h j +

(n

j

)⎛⎝ j∑

s=1

(−1) j−s−1βs−1(K )

⎞⎠ (8.3)

for 0 � j � n − 1, and h′′n = h′

n . The sum over an empty set is assumed zero.

8.3 Homology Spheres

Definition 8.1 K is called Cohen–Macaulay (over k), if H j (lkK I ; k) = 0 for anyI ∈ K and j < dim lkK I = n − 1 − |I |. If, moreover, Hn−1−|I |(lkK I ; k) ∼= k forany I ∈ K , then K is called Gorenstein* or (generalized) homology sphere.

The famous theorems of Reisner and Stanley (the reader is referred to the mono-graph (Stanley 1996)) tell that whenever K is Cohen–Macaulay (resp. Gorenstein*),its Stanley–Reisner algebra k[K ] is Cohen–Macaulay (resp. Gorenstein).

Given a characteristic function λ : [m] → V ∼= Rn we obtain a linear system of

parameters θ1, . . . , θn ∈ R[K ]. It generates an ideal which we denoted by Θ ⊂ R[K ]in Sect. 4.2. In Cohen–Macaulay case every linear system of parameters is a regularsequence. This implies (Stanley 1996):

dim(R[K ]/Θ)2 j = h j .

If K is a homology sphere, then R[K ] is Gorenstein. Thus its quotient by a linearsystem of parameters R[K ]/Θ is a Gorenstein algebra of Krull dimension zero. Thisimplies that R[K ]/Θ is a Poincare duality algebra (Meyer and Smith 2005, Part 1).

Now let Δ be a complete multi-fan based on a homology sphere K . We havethe ring epimorphism R[K ]/Θ → A∗(Δ) (see Remark 7.6). Since both algebras

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have Poincare duality, this map is an isomorphism (see Lemma 7.4). This proves thefollowing

Theorem 8.2 Let Δ be a complete multi-fan based on a homology sphere K . ThenA∗(Δ) ∼= R[K ]/Θ . It follows that dimA2 j (Δ) = h j , the h-number of K .

Note that Poincare duality implies the well-knownDehn–Sommerville relations forhomology spheres: h j = hn− j .

We are in position to prove Lemma 6.1 which states that the volume polynomial ofan elementary multi-fan Δ on the vectors λ(i) ∈ V (i = 1, . . . , n + 1), is equal, upto multiplicative constant, to (

∑n+1i=1 αi ci )n , where (α1, . . . , αn+1) is a linear relation

on λ(i)’s.

Proof of Lemma 6.1 The underlying simplicial complex ofΔ is the boundary of a sim-plex, which is a sphere. Therefore, by Theorem 8.2 we haveA∗(Δ) ∼= R[∂�[n+1]]/Θ .Hence the ideal Ann VΔ ⊂ R[∂1, . . . , ∂n+1] is generated by∏n+1

i=1 ∂i (Stanley–Reisnerrelation) and linear differential operators θ j = ∑n+1

i=1 λi, j∂i for j = 1, . . . , n. Here(λi, j )

nj=1 are the coordinates of the vector λ(i) for each i = 1, . . . , n + 1. Since∑n+1

i=1 αiλ(i) = 0 we have a linear relation∑n+1

i=1 αiλi, j = 0 for each j = 1, . . . , n.Now it is easy to check that the differential operators

∏n+1i=1 ∂i and θ j = ∑n+1

i=1 λi, j∂i ,j = 1, . . . , n annihilate the polynomial (α1c1 + · · · + αn+1cn+1)

n . Thus, accordingto Proposition 7.2, VΔ coincides with (α1c1 + · · · + αn+1cn+1)

n up to constant. ��

8.4 Homology Manifolds

Definition 8.3 K is called Buchsbaum (over k), if H j (lkK I ; k) = 0 for any I ∈ K ,I = ∅ and j < dim lkK I = n − 1 − |I |. If, moreover, Hn−1−|I |(lkK I ; k) ∼= k forany I ∈ K , I = ∅, then K is called a homology manifold. K is called an orientablehomology manifold if Hn−1(K ; Z) ∼= Z.

The difference from the Cohen–Macaulay case is that there are no restrictions onthe topology of K = lkK ∅ itself. Similar to Cohen–Macaulay property, the term“Buchsbaum” indicates that the corresponding algebra k[K ] is Buchsbaum (the resultof Schenzel 1981). In Buchsbaum case linear system of parameters is no longer aregular sequence. Nevertheless, Buchsbaum complexes are extensively studied. First,Schenzel’s theorem (Schenzel 1981) tells that if K is a Buchsbaum complex, then

dim(k[K ]/Θ)2 j = h′j

for j = 0, . . . , n and the h′-numbers determined by (8.2). Second, there is a theory ofsocles of Buchsbaum complexes introduced by Novik and Swartz (2009a) which webriefly review next.

Let M be a module over the graded polynomial ring k[m] := k[x1, . . . , xm]. Thesocle of M is the following subspace

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SocM := {a ∈ M | a · k[m]+ = 0}.

which is obviously a k[m]-submodule of M.If K is Buchsbaum, then there exists a submodule IN S ⊂ Soc(k[K ]/Θ) such that

(IN S)2 j ∼=(n

j

)H j−1(K ; k),

where the right hand sidemeans the direct sumof(nj

)copies of H j−1(K ; k).Moreover,

the result ofNovik andSwartz (2009b) tells thatwhenever K is an orientable connectedhomology manifold, then IN S coincides with Soc(k[K ]/Θ) and the quotient

(k[K ]/Θ)/I<2nN S

is a Gorenstein algebra (thus Poincare duality algebra). Here I<2nN S is the part of IN S

taken in all degrees except the top one, 2n. The definition of h′′-numbers (8.3) impliesthat

dim((k[K ]/Θ)/I<2nN S )2 j = h′′

j for 0 � j � n.

Now let Δ be a complete multi-fan based on an orientable connected homologymanifold K . Recall from Definition 7.3 that W ∗ denotes the subspace of H∗(Δ) =R[K ]/Θ whose graded components are

W 2 j ={a ∈ (R[K ]/Θ)2 j |

∫a · (R[K ]/Θ)2n−2 j = 0

}

={a ∈ (R[K ]/Θ)2 j |

∫a · R[m]2n−2 j = 0

}. (8.4)

By definition, A∗(Δ) = PD(R[K ]/Θ) = (R[K ]/Θ)/W ∗. The socle Soc(R[K ]/Θ)

lies in W ∗ in all degrees except the top one since it is killed by R[m]+. Therefore wehave a well-defined ring epimorphism

(R[K ]/Θ)/I<2nN S � A∗(Δ).

Again, since both algebras have Poincare duality, there holds

Theorem 8.4 Let Δ be a complete multi-fan based on oriented connected homologymanifold K . Then A∗(Δ) ∼= (R[K ]/Θ)/I<2n

N S . It follows that dimA2 j (Δ) = h′′j , the

h′′-number of K .

In this casePoincare duality implies thewell-knowngeneralizedDehn–Sommervillerelations for oriented homology manifolds: h′′

j = h′′n− j (see Novik and Swartz 2009a

and references therein).

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8.5 General Situation

Let Δ be an arbitrary complete multi-fan. In the largest generality we do not have acombinatorial description for the dimensions of graded components of the multi-fanalgebra.

Conjecture 8.5 Let wch be a simplicial cycle, λ : [m] → V a characteristicfunction, and Δ = (wch, λ) the corresponding complete multi-fan. The numbersd j = dimA2 j (Δ) do not depend on λ.

9 Geometry of Multi-Polytopes and Minkowski Relations

Here we give another proof of Theorem 8.4 which shows the geometrical nature of theelements lying in the socle of R[K ]/Θ when K is an oriented homology manifold. Itrelates on explicit computations in coordinates but reveals an interesting connectionwith the Minkowski type relations, appearing in convex geometry. Recall the basicMinkowski theorem on convex polytopes.

Theorem 1 (Minkowski)

1. (Direct) Let P be a convex full-dimensional polytope in euclidian space Rn. Let

V1, . . . , Vm be the (n − 1)-volumes of facets of P and n1, . . . ,nm be the outwardunit normal vectors to facets. Then

∑i Vini = 0 (the Minkowski relation).

2. (Inverse) Let n1, . . . ,nm be the vectors of unit lengths, spanning Rn, and let

V1, . . . , Vm be positive numbers satisfying the Minkowski relation. Then thereexists a convex polytope P whose facets have outward normal vectors ni andvolumes Vi . Such polytope is unique up to parallel shifts.

Usually only part (2) is called Minkowski theorem, since part (1) is fairly simple.The direct Minkowski theorem has a straightforward generalization.

Theorem 9.1 Let∑

s as Qs be a collection of k-dimensionalmulti-polytopes in euclid-ian space R

n, forming a closed orientable cycle. Let Vol(Qs) be the k-volume, andνs ∈ Λn−k

Rn be the unit normal skew form of the multi-polytope Qs. Then there holds

a relation∑

s as Vol(Qs)νs = 0 in Λn−kRn.

In the next subsection we explain the precise meaning of the terms used in thestatement and give the proof.

9.1 Cycles of Multi-Polytopes

As before, V ∗ ∼= Rn denotes the ambient affine space of n-dimensional polytopes,

coming with fixed orientation. Let k � n and let Π be an oriented k-dimensionalaffine subspace of V ∗. Let Q be a k-dimensional multi-polytope in Π . Then Q willbe called a k-dimensional multi-polytope in V ∗.

First let k > 0. Denote by GMPk the group (or a vector space over R) freelygenerated by all k-dimensional multi-polytopes in V ∗, where we identify the element

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Q (i.e. Q with reversed orientation of the underlying subspace) and −Q. If k = 0, themulti-polytope is just a point with weight. In this case let GMP0 denote the group offormal sums of points whose weights sum to zero. Formally set GMP−1 = 0. Definethe differential d : GMPk → GMPk−1 by setting

dQ :=∑

Fi : facet of QFi ,

and extending by linearity. Note that each facet comes with the canonical orientation:we say that the hyperplane Hi containing Fi is positively oriented if

(a positive basis of Hi , λ(i))

is a positive basis of V . Thus the expression above is well defined.

Definition 9.2 An element A = ∑s as Qs ∈ GMPk which satisfies d A = 0 is called

a cycle of k-dimensional multi-polytopes.

As in Sect. 5, assume that there is a fixed inner product in V . This allows to definethe inner product on the skew forms. In particular, ifΠ is an oriented affine k-subspacein V ∗ ∼= V , we may define its unit normal skew form νΠ ∈ Λn−kV as the uniqueelement of Λn−kΠ⊥ ∼= R which corresponds to the positive orientation of Π⊥ andsatisfies ‖νΠ‖ = 1. It is easy to see that if dimΠ = n − 1, the form νΠ is just thepositive unit normal vector to Π .

Let us prove Theorem 9.1.

Proof The idea of proof is straightforward and quite similar to the proof of classicalMinkowski theorem: at first we prove the case k = n, then reduce the general case tothe case k = n by projecting

∑s as Vol(Qs)νs to all possible k-subspaces. Note that

the case n = 0 should be treated separately, but in this case the statement is trivial.(1) Suppose k = n. Then all multi-polytopes Qs are full-dimensional. Their

underlying subspaces Πs coincide with V up to orientation. Without loss of gen-erality assume that all orientations coincide with that of V . Normal skew forms lie inΛ0V ∼= R and are equal to 1. Hence we need to prove that

∑as Vol(Qs) = 0 for

any cycle of n-multi-polytopes. Recall the wall-crossing formula (Hattori andMasuda2003, Lemma 5.3):

Lemma 9.3 Let P be a multi-polytope and H = Hi be one of the supporting hyper-planes: H = Hi . Let uα and uβ be elements in V ∗\⋃m

i=1 Hi such that the segmentfrom uα to uβ intersects the wall H transversely atμ, and does not intersect any otherHj = H. Then

DHP (uα) − DHP (uβ) =∑

i :Hi=H

sgn〈uβ − uα, λ(i)〉DHFi (μ),

where Fi is the facet of P, andDHFi : Hi → R is its Duistermaat–Heckman function.

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Consider a cycle of multi-polytopes A = ∑ls=1 asQs . Let H denote the set of all

supporting hyperplanes of all polytopes Qs , s = 1, . . . , l. We have a function

DHA : V ∗\⋃H∈H

H → R, DHA :=l∑

s=1

as DHQs .

Let us choose a hyperplane H ∈ H and two points uα and uβ in V ∗\⋃H∈H Hsuch that the segment from uα to uβ intersects the wall H transversely at μ and doesnot intersect any other wall fromH. Let us sum the differences DHP (uα)−DHP (uβ)

taken with coefficients as over all multi-polytopes Qs for which H is a supportinghyperplane. Since d A = 0, Lemma 9.3 implies that this sum is zero. Obviously, thissum equals DHA(uα) − DHA(uβ).

This argument shows that crossing of any wall does not change the value of DHA.Therefore, DHA is constant (where it is defined). Since DHA has compact support, itmust be constantly zero. Thus

∑as Vol(Qs) =

∫V ∗

DHA = 0.

(2) Let us prove the theorem for general k. Consider a generic oriented k-subspaceΠ ⊂ V ∗ and let ν ∈ Λn−kV ∗ be its normal skew form. Let Γ : V ∗ → Π be theorthogonal projection. Then the image of Qs under Γ is a full-dimensional multi-polytope in Π , which we denote by Γ (Qs). The sum

∑ls=1 asΓ (Qs) is a cycle of

k-dimensional multi-polytopes in Π . Therefore, step (1) implies

l∑s=1

as Vol(Γ (Qs)) = 0.

By the standard property of orthogonal projections we have

Vol(Γ (Qs)) = Vol(Qs) · 〈νs, ν〉.

Hence ⟨l∑

s=1

as Vol(Qs)νs, ν

⟩= 0,

and this holds for any generic skew form ν. Thus∑l

s=1 as Vol(Qs)νs = 0 which wasto be proved. ��

9.2 Relations inA∗(Δ) as Minkowski Relations

Let K be an oriented homology (n − 1)-manifold and Δ be a multi-fan based on K .Suppose that every simplex I ∈ K is oriented somehow. This defines an orientationof each subspace HI = ⋂

i∈I Hi (for example, by the rule “positive orientation of

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H′′I ⊕λ(i1)⊕· · ·⊕λ(ik) is a positive orientation of V if (i1, . . . , ik) is a positive order

of vertices of I ). Recall from Sect. 5 that λ(I ) denotes the skew form∧

i∈I λ(i) andcovol(I ) = ‖λ(I )‖. Consider an arbitrary skew form μ ∈ ΛkV ∗ and let

λ(I )μ := 〈λ(I ), μ〉.

Let Ck(K ; R), 0 � k � n − 1 denote the group of cochains on K andδ : Ck(K ; R) → Ck+1(K ; R) be the standard cochain differential. We also need toaugment the cochain complex in the top degree, sowe formally setCn(K ; R) := R andlet δ : Cn−1(K ; R) → Cn(K ; R) be the evaluation of a cochain on the fundamentalchain of K .

An element a ∈ Ck−1(K ; R), k � n will be called a (coaugmented) cocycle ifδa = 0. Then, since K is an oriented manifold, the Poincare dual

∑I :|I |=k a(I )FI of

a is a cycle of (n − k)-dimensional multi-polytopes in V ∗. (Notice that in the casek = n we get a formal sum of points whose weights sum is zero. If we do not requirethat a is coaugmented, then we do not get a cycle of 0-dimensional multi-polytopes).

Proposition 9.4 For any coaugmented cocycle a ∈ Ck−1(K ; R) and any μ ∈ ΛkV ∗there exists a relation ∑

I :|I |=k

a(I )λ(I )μ∂I = 0

in A∗(Δ) ∼= (R[K ]/Θ)/I<2nN S .

Proof Let us apply∑

I :|I |=k a(I )λ(I )μ∂I to the volume polynomial VΔ and evaluatethe result at a point c = (c1, . . . , cm):

∑I :|I |=k

a(I )λ(I )μ∂I VΔ|c =∑

I :|I |=k

a(I )λ(I )μVol(FI )

covol(I ).

Here we used Lemma 5.1. Note that the skew form λ(I )/ covol(I ) is by definitiona unit normal skew form to the ambient subspace of a multi-polytope FI . Since∑

I :|I |=k a(I )FI is a cycle of multi-polytopes, Theorem 9.1 implies

∑I :|I |=k

a(I )Vol(FI )λ(I )

covol(I )= 0.

Taking inner product with μ implies

∑I :|I |=k

a(I )λ(I )μVol(FI )

covol(I )= 0.

Hence the polynomial∑

I :|I |=k a(I )λ(I )μ∂I VΔ evaluates to zero at any point c. There-fore it vanishes as a polynomial. Thus

∑I :|I |=k a(I )λ(I )μ∂I ∈ Ann VΔ which proves

the statement. ��

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We see that Minkowski theorem allows to construct linear relations in A∗(Δ).Actually these relations exhaust all relations in A∗(Δ). Let us state the result ofAyzenberg (2016) in terms of Minkowski relations:

Proposition 9.5 (Ayzenberg 2016) Let K be an oriented homology manifold.

1. There is an isomorphism of vector spaces

(R[K ]/Θ)2k ∼= 〈xI | I ∈ K , |I | = k〉/⟨ ∑

I :|I |=k

a(I )λ(I )μxI

⟩

where a runs over all exact (k − 1)-cochains on K and μ runs over μ ∈ ΛkV ∗.2. There is an isomorphism of vector spaces

((R[K ]/Θ)/I<2nN S )2k ∼= 〈xI | I ∈ K , |I | = k〉

/⟨ ∑I :|I |=k

a(I )λ(I )μxI

⟩

where a runs over all coaugmented closed (k−1)-cochains on K andμ runs overμ ∈ ΛkV ∗.

Recall that (IN S)2k ∼= (nk

)Hk−1(K ; R). From Proposition 9.5 it can be seen that the

difference between the vector spaces R[K ]/Θ and (R[K ]/Θ)/I<2nN S arises from the

difference between closed cochains on K and exact cochains. This explains how thecohomology Hk−1(K ) appears in the description of IN S . Themultiple

(nk

)comes from

the choice of the skew form μ ∈ ΛkV on which the Minkowski relation is projected.

Problem 9.6 Let Δ be a general complete simplicial multi-fan. Is it true that A∗(Δ)

is isomorphic, as a vector space, to the quotient of 〈xI | I ∈ K 〉 by linear relationsarising from Minkowski relations? What are these Minkowski relations?

9.3 Inverse Minkowski Theorem

It is tempting to formulate and prove the inverse Minkowski theorem for multi-polytopes. First, we need to modify the statement. The original formulation tells thatthere exists a convex polytope with the given normal vectors and the volumes of facets,but it tells nothing about the combinatorics of the polytope.Wemay ask amore specificquestion, namely

Question 9.7 For a given complete simplicial multi-fan Δ with the rays generated byunit vectors n1, . . . ,nm, and a given m-tuple of real numbers V1, . . . , Vm satisfying∑

Vini = 0, does there exist a multi-polytope based on Δ whose facets have (n− 1)-volumes V1, . . . , Vm? If yes, is it unique?

A simple example shows that the answer, even for the question of existence, maybe negative.

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Example 9.8 Let Δ be the normal fan of a 3-dimensional cube. Δ is an ordinaryfan supported by a simplicial complex K , which is the boundary of an octahe-dron. Let {1, 2, 3,−1,−2,−3} be the set of vertices of K and λ(±1) = (±1, 0, 0),λ(±2) = (0,±1, 0), λ(±3) = (0, 0,±1) be the generators of the corresponding raysofΔ (see Example 6.6). The multi-polytopes based onΔ are the bricks with sides par-allel to coordinate axes. Minkowski relations can be written as Vol(Fi ) = Vol(F−i )

for i = 1, 2, 3. Let us take the numbers V±1 = 0, V±2 = V±3 = 1. These num-bers satisfy Minkowski relations, but there are no bricks whose facets have volumesV±1, V±2, V±3. Indeed, V±1 = 0 implies that one of the sides of a brick has length 0,but this would imply that either V±2 = 0 or V±3 = 0.

Nevertheless, the answer to Question 9.7 is completely controlled by the multi-fanalgebra. Recall thatA∗(Δ)may be interpreted as the algebra of differential operatorsDup toAnn(VΔ). Therefore, for every a ∈ A2 j (Δ), there is awell-defined homogeneouspolynomial aVΔ of degree n− j . In particular, each element a ∈ A2n−2(Δ) determinesa linear homogeneous polynomial aVΔ = V1c1 +· · ·+Vmcm ∈ R[c1, . . . , cm]1. Thislinear polynomial is annihilated by θ j = ∑

i∈[m] λi, j∂ j ∈ Ann VΔ, j = 1, . . . , n, seeLemma 5.1 or Remark 7.6. This means

∑i∈[m]

Viλ(i) = 0, (9.1)

which can be considered as aMinkowski relation. Let Mink denote the vector space ofall m-tuples (V1, . . . , Vm) satisfying (9.1). Thus we obtain a natural monomorphismη : A2n−2(Δ) → Mink, a → (V1, . . . , Vm), where aVΔ = V1c1 + · · · + Vmcm .

Theorem 9.9 Let Δ be a complete simplicial multi-fan with characteristic functionλ and assume that |λ(i)| = 1 for each i ∈ [m]. Let V = (V1, . . . , Vm) ∈ Mink.Let P ∈ Poly(Δ) be a multi-polytope and ∂P = c1∂1 + · · · + cm∂m ∈ A2(Δ) be itsfirst Chern class. Then the polytope P has facet volumes V1, . . . , Vm if and only ifη(∂n−1

P ) = (n − 1)!V .

Proof Assume that η(∂n−1P ) = (n − 1)!V . Then ∂n−1

P VΔ = (n − 1)!(V1c1 + · · · +Vmcm). Hence ∂i∂

n−1P VΔ = (n − 1)!Vi for i ∈ [m]. On the other hand, Corollary 5.7

implies

∂i∂n−1P VΔ =(n − 1)!Vol(Fi )/ covol(i)=(n − 1)!Vol(Fi )/|λ(i)| = (n − 1)!Vol(Fi ).

Thus Vi = Vol(Fi ). The opposite direction is proved similarly. ��Note that dimMink = m − n.

Corollary 9.10 Existence in Question 9.7 holds for a given multi-fan Δ and all m-tuples (V1, . . . , Vm) ∈ Mink if and only if the following two conditions hold:1. dimA2(Δ) = dimA2n−2(Δ) = m − n;2. the power map A2(Δ) → A2n−2(Δ), ∂ → ∂n−1 is surjective.

Uniqueness holds if the power map is bijective.

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Remark 9.11 Note that even the condition dimA2(Δ) = m − n may fail to hold. Asan example, consider a multi-fan having a ghost vertex, say 1. As in general, we haven relations θ1, . . . , θn , lying in the kernel of the linear map 〈∂1, . . . , ∂m〉 � A2(Δ).But the element ∂1, corresponding to the ghost vertex, also vanishes in A2(Δ). ThusdimA2(Δ) < m − n.

There exist more nontrivial examples. For example, if the underlying simplicialcomplex K is disconnected,with connected components K1, . . . , Kr on disjoint vertexsets [m1], . . . , [mr ], r > 1, then each connected component contributes at mostm1−nin the total dimension ofA2(Δ) (see the operation of connected sumofPoincare dualityalgebras introduced in Sect. 11.1). Thus in the disconnected case dimA2(Δ) � m−rn,where m = m1 + · · · + mr . Nevertheless, the inverse Minkowski theorem can berefined in an obvious way: we should considerMinkowski relations on each connectedcomponent.

Remark 9.12 The power map A2(Δ) → A2n−2(Δ) is a polynomial map of degreen−1 between real vector spaces of equal dimensions. It is a complicated object whichmay be interesting on its own. One of the consequences from Corollary 9.10 is thatthe existence in the inverse Minkowski theorem holds for a multi-fan Δ wheneverdimA2(Δ) = m − n and n is even.

10 Recognizing Volume Polynomials and Multi-Fan Algebras

A natural question is: which homogeneous polynomials are the volume polynomials,and which Poincare duality algebras appear as A∗(Δ)? The answer to the secondquestion seems quite unexpected.

Theorem 10.1 For every Poincare duality algebra A∗ generated in degree 2 thereexists a complete simplicial multi-fan Δ such that A∗ ∼= A∗(Δ).

Recall that the symmetric array of nonnegative integers (d0, d1, . . . , dn), d j =dn− j , is called unimodal, if

d0 � d1 � · · · � d�n/2�.

Corollary 10.2 There exist multi-fans Δ, for which the array

(dimA0(Δ), dimA2(Δ), . . . , dimA2n(Δ))

is not unimodal.

Proof An example of Poincare duality algebra generated in degree 2, for whichdimensions of graded components are not unimodal was given by Stanley (1978).Theorem 10.1 implies that there exists a multi-fan, which produces this algebra. ��

The construction of the volume polynomial is linear with respect to weights. LetMultiFansλ denote the vector space of all multi-fans with the given characteristicfunction λ : [m] → V . Then we obtain a linear map

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Ωλ : MultiFansλ → R[c1, . . . , cm]n, (10.1)

which maps Δ to its volume polynomial VΔ.Before giving the proof of Theorem 10.1 we characterize volume polynomials of

general position, in the sense explained below. For this goal we study the propertiesof the map Ωλ.

10.1 Characterization of Volume Polynomials in General Position

There is a necessary condition on VΔ. If λ is a characteristic function and θ j =∑i∈[m] λi, j∂i ∈ 〈∂1, . . . , ∂m〉, j = 1, . . . , n are the corresponding linear forms, then

θ j VΔ = 0, see Remark 7.6. Thus the subspace

Ann2 VΔ = {D ∈ 〈∂1, . . . , ∂m〉 | DVΔ = 0}

has dimension at least n. It happens that in most situations this is also a sufficientcondition for a polynomial to be a volume polynomial.

At first let us consider the situation of general position to demonstrate the argument.Assume that all characteristic vectors λ(1), . . . , λ(m) ∈ R

n are in general position,whichmeans that every n of them are linearly independent. Given a fixed characteristicfunction λ : [m] → V in general position, we may pick up any simplicial cyclewch ∈ Z(�(n−1)

[m] ; R), consider a complete multi-fan Δ = (w, λ) and take its volumepolynomial. This defines a map which we previously denoted by Ωλ:

Ωλ : Z(�(n−1)[m] ; R) → R[c1, . . . , cm]n,

from the (n−1)-simplicial cycles onm vertices to homogeneous polynomials of degreen. This map is linear and injective by Corollary 5.5. As before, let θ j , j = 1, . . . , nbe the linear differential operators associated with λ (i.e. a basis of the image of themap λ� : V ∗ → (Rm)∗). Let

Annn Θ = {Ψ ∈ R[c1, . . . , cm]n | θ jΨ = 0 for each j = 1, . . . , n}

be the vector subspace of polynomials annihilated by differential operators Θ =(θ1, . . . , θn). As we have seen, ifΔ has a characteristic function λ, then VΔ ∈ Annn Θ .Thus the image of Ωλ lies in Annn Θ .

Lemma 10.3 If λ is in general position, then Ωλ : Zn−1(�(n−1)[m] ; R) → Annn Θ is

an isomorphism.

Proof Let us compute the dimensions of the domain and the target. There are non-simplices in �(n−1)

[m] , thus Zn−1(�(n−1)[m] ) = Hn−1(�(n−1)

[m] ). All Betti numbers of

�(n−1)[m] between the top and the bottom vanish, thus via Euler characteristic we get

dim Hn−1(�(n−1)[m] )=

(m

n

)−(

m

n − 1

)+(

m

n − 2

)− · · · + (−1)n

(m

0

). (10.2)

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Now let us compute dimAnnn Θ . Take a linear change of variables c1, . . . , cm �c′1, . . . , c

′m such that θ j becomes the partial derivative ∂

∂c′jfor j = 1, . . . , n. Thus,

after the change of variables, Annn Θ becomes the set {Ψ ∈ R[c′1, . . . , c

′m]n | ∂

∂c′jΨ =

0, j = 1, . . . , n} which is the same as R[c′n+1, . . . , c

′m]n . Thus dimAnnn Θ =(m−n+n−1

n

) = (m−1n

). This number coincides with (10.2). ��

Let Gm,n denote the Grassmann manifold of all (unoriented) n-planes in (Rm)∗.We can introduce the standard Plücker coordinates on Gm,n . If

{θ j =

m∑i=1

λi, j xi

}

j=1,...,n

(10.3)

is a basis in L ∈ Gm,n , then the Plücker coordinates of L are all maximal minors ofthe m × n matrix (λi, j ).

Any n-plane L ∈ Gm,n determines an m-tuple of vectors in V ∼= Rn as follows:

the basis (10.3) determines the tuple {λ(i) = (λi,1, . . . , λi,n)}i∈[m]. The base changein L induces the natural action of GL(n, R) on the m-tuples. By abuse of terminologywe call λ : [m] → V the characteristic function corresponding to L ∈ Gm,n althoughthis characteristic function is determined only up to automorphism of V .

Proposition 10.4 Let Ψ ∈ R[c1, . . . , cm]n be a homogeneous polynomial. Supposethat the vector subspace Ann2 Ψ = {D ∈ 〈∂1, . . . , ∂m〉 | DΨ = 0} contains ann-dimensional subspace L ∈ Gm,n with all Plücker coordinates non-zero. Then Ψ isa volume polynomial of some multi-fan.

Proof Let us pick a basis {θ j = ∑i λi, j xi } j=1,...,n in L arbitrarily. Non-vanishing of

all Plücker coordinates means that the corresponding characteristic function λ is ingeneral position. By assumption, Ψ ∈ Annn Θ . Thus Ψ is a volume polynomial ofsome multi-fan based on λ according to Lemma 10.3. ��

10.2 Proof of Theorem 10.1

LetA∗ be an arbitrary Poincare duality algebra over R generated byA2. Let 2n be theformal dimension ofA and p = dimA2. Take any p+n elements x1, . . . , xp+n ∈ A2

in general position (i.e. every p of them are linearly independent). There are n linearrelations on x1, . . . , xp+n inA2 of the form

∑i λi, j xi = 0, j = 1, . . . , n. Since xi are

in general position, every maximal minor of the (p+ n)× n-matrix |λi, j | is non-zero.As in the proof of Proposition 7.2, consider the polynomial

ΨA(c1, . . . , cm) = 1

n!∫

(c1x1 + · · · + cmxm)n,

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where∫ : A2n

∼=→ R is any isomorphism. The linear differential operator θ j =∑i λi, j∂i annihilates ΨA for j = 1, . . . , n. Indeed:

(m∑i=1

λi, j∂

∂ci

)1

n!∫

(c1x1 + · · · + cmxm)n

= 1

n! · n∫ (

m∑i=1

λi, j xi

)(c1x1 + · · · + cmxm)n−1 = 0.

Since θ j are in general position, Proposition 10.4 implies that ΨA = VΔ for somemulti-fan Δ. Therefore the corresponding Poincare duality algebras A∗ and A∗(Δ)

are isomorphic by Proposition 7.2.

10.3 Non-General Position

Nowwewant to studywhich polynomials are volumepolynomialswithout the assump-tion of general position.

Let I ⊂ [m] and let αI : RI → R

m be the inclusion of the coordinate subspace.Then α∗

I : (Rm)∗ = 〈∂1, . . . , ∂m〉 → (RI )∗ is a projection map. For a linear subspaceΠ ⊂ (Rm)∗ of dimension at least n consider the following collection of subsets of[m]:

dep(Π) := {I ⊂ [m] | |I | � n and α∗I |Π : Π → (RI )∗ is not surjective}.

Lemma 10.5 Let Π ⊂ (Rm)∗ and dimΠ � n. Then there exists a subspace L ⊂ Π

such that dim L = n and dep(L) = dep(Π).

Proof When dimΠ = n the statement is trivial so we assume dimΠ > n. The prooffollows from the general position argument. If ϕ : Π → U is an epimorphism, anddimΠ > n � dimU , then the set of all n-planes in Π which map surjectively toU isa complement to a subvariety of positive codimension inside the set of all n-subspacesof Π . This argument applied to all maps α∗

I |Π : Π → (RI )∗ proves that any genericn-plane L in Π satisfies dep(L) = dep(Π). ��

Let Ψ be a homogeneous polynomial of degree n and Ann2 Ψ ⊂ 〈∂1, . . . , ∂m〉 =(Rm)∗ be its annihilator subspace.

Theorem 10.6 A homogeneous polynomial Ψ ∈ R[c1, . . . , cm]n is a volume poly-nomial of some complete simplicial multi-fan if and only if the following conditionshold:1. dim Ann2 Ψ � n,

2. ∂IΨ = 0 whenever I ∈ dep(Ann2 Ψ ).

Proof The necessity of these conditions is already proved. Indeed, the first conditionfollows from the fact that Ann2 VΔ contains the image of λ� : V ∗ → (Rm)∗ =

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〈∂1, . . . , ∂m〉 which has dimension n, see Remark 7.6. If I ∈ dep(Ann2 VΔ), then∗-condition (see Sect. 2.2) implies I /∈ K , and therefore ∂I VΔ = 0 by Lemma 5.1.

Let us prove sufficiency. By Lemma 10.5 we may choose an n-dimensionalplane L ⊂ Ann2 Ψ such that dep(L) = dep(Ann2 Ψ ). Therefore, by assumption,I ∈ dep(L) implies ∂IΨ = 0. Let λ : [m] → V be the characteristic function corre-sponding to L ∈ Gm,n . The condition I ∈ dep(L) is equivalent to the condition thatvectors {λ(i)}i∈I are linearly dependent.

Consider a simplicial complex Matrλ determined by the condition: {i1, . . . , ik} ∈Matrλ if and only if λ(i1), . . . , λ(ik) are linearly independent. Thus Matrλ =2[m]\dep(L). In a sense, the complex Matrλ can be considered as a maximal sim-plicial complex on [m] for which λ is a characteristic function (this construction issimilar to the universal complexes introduced in Davis and Januszkiewicz (1991)).

We have I /∈ Matrλ if and only if Θ → 〈∂i 〉i∈I is not surjective. It is easily seenthat multi-fans having characteristic function λ are encoded by the simplicial (n− 1)-cycles onMatrλ. As before, we have amapΩλ : Zn−1(Matrλ; R) → R[c1, . . . , cm]nwhich associates a volume polynomial VΔ with a multi-fan Δ = (wch, λ) for wch ∈Zn−1(Matrλ; R). Let

Annn(L , {∂I }I∈dep(L))

denote the subspace of all homogeneous polynomials of degree nwhich are annihilatedby linear differential operators from L and by the products ∂I , I ∈ dep(L) (⇔ I /∈Matrλ). We already proved that the image of Ωλ lies in Annn(L , {∂I }I∈dep(L)). Weneed to prove that the map

Ωλ : Zn−1(Matrλ; R) → Annn(L , {∂I }I∈dep(L))

is surjective. Since Ωλ is injective, it is enough to show that dimensions of the twospaces are equal.

First of all notice that Matrλ is by construction the underlying simplicial complexof a linear matroid. Hence Matrλ is a Cohen–Macaulay complex of dimension n − 1(see e.g. Stanley 1977). The number dim Zn−1(Matrλ; R) = dim Hn−1(Matrλ; R)

coincides with the top h-number hn(Matrλ) of the Cohen–Macaulay complex Matrλ.Consider the Stanley–Reisner ring R[Matrλ] = R[∂1, . . . , ∂m]/(∂I | I /∈ Matrλ),

and its quotient by a linear system of parameters L ⊂ 〈∂1, . . . , ∂m〉:

R[Matrλ]/(L) = R[∂1, . . . , ∂m]/(L , {∂I }I∈dep(L))

Claim 10.7 dim(R[Matrλ]/(L))2n = dimAnnn(L , {∂I }I∈dep(L)).

This follows from basic linear algebra. There is a non-degenerate pairing

R[∂1, . . . , ∂m]2n ⊗ R[c1, . . . , cm]n → R.

For any subspace U ⊂ R[∂1, . . . , ∂m]2n we have dimR[∂1, . . . , ∂m]2n/U =dimU⊥. Taking the degree 2n part of the ideal (L , {∂I }I∈dep(L)) as U proves theclaim.

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Now, since Matrλ is Cohen–Macaulay, the socle of R[Matrλ]/(L) coincides with(R[Matrλ]/(L))2n . On the other hand, this dimension equals hn(Matrλ), the top h-number, as follows from the basic theory of Stanley–Reisner rings (Stanley 1996). Wehave

dimAnnn(L , {∂I }I∈dep(L)) = dim SocR[Matrλ]/(L)2n = dim Hn−1(Matrλ; R)

which finishes the proof of the theorem. ��Remark 10.8 Lemma 10.3 describing the general position is a particular case of The-orem 10.6. In the case of general position the matroid complex Matrλ is just the(n − 1)-skeleton of a simplex on m vertices.

10.4 Global Structure of the Set of Multi-Fans

LetGnm denote theGrassmannian of all codimensionn planes inR

m ∼= R[c1, . . . , cm]1.Obviously,Gm,n can be identified withGn

m by assigning L⊥ ⊂ Rm to L ⊂ (Rm)∗. We

have already seen, that characteristic function λ : Rm → V determines the element

L ∈ Gm,n defined as the image of λ� : V ∗ → (Rm)∗. The corresponding element ofGn

m is the subspace Y = L⊥ = Ker λ ⊂ Rm .

Let SkY denote the k-th symmetric power of Y ∈ Gnm , so we have S

kY ⊂ SkRm =R[c1, . . . , cm]k . We have

SkY ⊂ Annk Y⊥ = {Ψ ∈ R[c1, . . . , cm]k | DΨ = 0 for any D ∈ Y⊥},

and both spaces have dimension(m−n+k−1

k

). This implies that the vector bundle

{(Y, Ψ ) ∈ Gnm × R[c1, . . . , cm]k | Ψ ∈ Annk Y⊥} → Gn

m

is Skγ , the k-th symmetric power of the canonical bundle γ over Gnm . We denote its

total space by E(Skγ ).Consider the set of all characteristic functions in V up to linear automorphism of

V :

CharFunc := {λ : Rm → V | Im λ = V }/GL(V ).

Let MultiFans denote the set of all complete simplicial multi-fans on the set [m](considered up to automorphisms of V again).We have amapMultiFans → CharFuncwhich maps a multi-fan to its characteristic function. The fiber of this map over λ isthe vector space MultiFansλ ∼= Zn−1(Matrλ; R) introduced earlier.

We have a commutative square

MultiFans ��

��

E(Snγ )

Snγ��

CharFunc∼= �� Gn

m

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The lower map associates a codimension n subspace Y = Ker λ to a characteristicfunction λ. The upper map associates a volume polynomial to a multi-fan. The uppermap is linear on each fiber. The subset of characteristic functions in general positionmaps isomorphically to the subset of Gn

m with non-zero Plücker coordinates; the fiberover a generic pointmaps isomorphically according to Lemma 10.3. Exceptional fibersmap injectively and their images are described by Theorem 10.6.

11 Surgery of Multi-Fans and Algebras

In this section we study the behavior of the dimensions d j = dimA2 j (Δ) underconnected sums and flips of multi-fans.

11.1 Connected Sums

Recall that A∗(Ψ ) = R[∂1, . . . , ∂m]/Ann(Ψ ) denotes the Poincare duality algebraassociated with the homogeneous polynomial Ψ . For a graded algebra (or a gradedvector space) A∗ = ⊕

j A2 j let Hilb(A∗; t) = ∑

j (dim A j )t j denote its Hilbert func-

tion. Sometimes it will be convenient to use the notation dm(A∗) := (d0, d1, . . . , dn),where d j = dim A2 j , and dm(Δ) := dm(A∗(Δ)).

Let A1#A2 denote a connected sum of two Poincare duality algebras of the sameformal dimension 2n. By definition, A1#A2 = A1 ⊕ A2/ ∼ where ∼ identifies A0

1with A0

2 and A2n1 with A2n

2 . Actually, there is an ambiguity in the choice of the latteridentification, so in fact there exists a 1-dimensional family of connected sums of thegiven two algebras.We prefer to ignore this ambiguity in the following (the statementshold for any representative in the family).

We have dm(A1#A2) = dm(A1) + dm(A2) − (1, 0, . . . , 0, 1).

Lemma 11.1 Let Ψ1 ∈ R[c1, . . . , cm]n, Ψ2 ∈ R[c′1, . . . , c

′m′ ]n be the polynomials in

distinct variables. Then A∗(Ψ1 + Ψ2) ∼= A∗(Ψ1)#A∗(Ψ2).

Proof The mixed differential operators ∂i∂′i ′ vanish on Ψ1 + Ψ2, while ∂I (Ψ1 + Ψ2)

equals ∂I (Ψ1) (resp. ∂I (Ψ2)) if I ⊂ [m] (resp. I ⊂ [m′]). ��LetΔ1,Δ2 be twomulti-fans, whose vertex sets are [m] = {1, . . . , n, n+1, . . . ,m}

and [m] = {1, . . . , n, n + 1, . . . , m} respectively, and let I denote the set of commonvertices: I = {1, . . . , n}. Assume that the weight of I is non-zero in both multi-fans and assume that characteristic functions of Δ1 and Δ2 coincide on I . Then wemay consider Δ1 and Δ2 as multi-fans with vertices [m] ∪I [m] = {1, . . . , n, n +1, . . . ,m, n + 1, . . . , m} and a common characteristic function. In this case we callthe cone-wise sum Δ1 + Δ2 a connected sum and denote it by Δ1#IΔ2 or simplyΔ1#Δ2.

Remark 11.2 It would be natural to assume that w1(I ) = −w2(I ), so that the conespanned by I contracts in the connected sum. This is consistent with the geometricalunderstanding how “the connected sum” should look like. However, we do not needthis assumption in the following proposition.

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Proposition 11.3 For a connected sum Δ1#Δ2 there holds

A∗(Δ1#Δ2) ∼= A∗(Δ1)#A∗(Δ2),

so that dm(Δ1#Δ2) = dm(Δ1) + dm(Δ2) − (1, 0, . . . , 0, 1).

Proof We need a technical lemma

Lemma 11.4 Let Δ be a multi-fan and I ⊂ [m] be a vertex set such that the cor-responding characteristic vectors {λ(i)}i∈I are linearly independent. Let VΔ be thevolume polynomial and VΔ\I ∈ R[ci | i ∈ [m]\I ]n be the homogeneous polyno-mial obtained by specializing ci = 0 in VΔ for each i ∈ I . Then A∗(VΔ\I ) ∼=A∗(VΔ)(=A∗(Δ)) as Poincare duality algebras.

Proof Using linear relations θ j = ∑mi=1 λi, j∂i = 0 in A2(Δ) we can exclude the

variables ∂i for i ∈ I . This proves that the set {∂i }i∈[m]\I spans A2(Δ). Therefore thepolynomial

VΔ\I = 1

n!∫

Δ

⎛⎝ ∑

i∈[m]\Ici xi

⎞⎠

n

determines the same Poincare duality algebra as VΔ. ��By the lemma we haveA∗(Δ1) ∼= A∗(VΔ1\I ) andA∗(Δ2) ∼= A∗(VΔ2\I ). Polyno-

mials VΔ1\I and VΔ2\I have distinct variables, thus

A∗(VΔ1\I + VΔ2\I ) ∼= A∗(VΔ1\I )#A∗(VΔ2\I )

according to Lemma 11.1. It remains to note that VΔ1\I + VΔ2\I is the result ofspecializing ci = 0 for i ∈ I in the polynomial VΔ1 + VΔ2 . Finally, we have

A∗(Δ1#Δ2) = A∗(VΔ1#Δ2)∼= A∗(VΔ1#Δ2\I )

∼= A∗(VΔ1\I + VΔ2\I ) ∼= A∗(Δ1)#A∗(Δ2).

��Remark 11.5 In the proof we actually did not need that I forms a cone of Δ1 or Δ.We only need that the values of characteristic function in vertices of I are linearlyindependent. Therefore, the connected summaybe defined in thismore general setting.

11.2 Flips

In this section we assume that Δ is based on an oriented pseudomanifold K . Our goalis to define a flip in a multi-fan. Consider separately two situations.

(1) Flips changing the number of vertices. Let us take a maximal simplex I ∈ K ,|I | = n. Let Flip1I (K ) be a simplicial complex whose maximal simplices are the same

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as in K except that we substitute I by Cone ∂ I . This operation adds the new vertexi , the apex of the cone. If λ : [m] → R

n is a characteristic function on K , we extendit to the set [m] � {i} by adding new value λ(i) such that the result is a characteristicfunction on Flip1I (K ). This defines an operation on multi-fans which we call the flipof type (1, n).

The inverse operation will be denoted Flipni . It is applicable to Δ if lkK i is iso-morphic to the boundary of a simplex and λ(Vert(lkK i)) is a linearly independent set.The inverse operation will be called the flip of type (n, 1).

(2) Flips preserving the set of vertices. Let S be a subset of Vert(K ) of cardinalityn+1 such that the induced subcomplex KS on the set S is isomorphic to ∂Δp−1∗Δq−1

with p + q = n + 1, p, q � 2. Let FlippS (K ) be the simplicial complex whosemaximal simplices are the same as in K away from S, and ∂Δp−1 ∗ Δq−1 is replacedby Δp−1 ∗ ∂Δq−1. If the set of vectors λ(S) is in general position, then this operationis defined on multi-fans. We call it the flip of type (p, q). It is easily seen that flips oftypes (p, q) and (q, p) are inverse to each other.

Of course (1, n)- and (n, 1)-flips can be viewed as particular cases of this construc-tion if we allow ghost vertices and formally set ∂Δ0 to be such a ghost vertex.

The following proposition tells that dimension vectors of multi-fan algebras changeunder the flips similar to h-vectors of simplicial complexes.

Theorem 11.6 Let Δ′ be a multi-fan obtained from Δ by a (p, q)-flip, p, q � 1.Then

dm(Δ′) − dm(Δ) = (1, . . . , 1︸︷︷︸q

, 0, . . . , 0) − (1, . . . , 1︸︷︷︸p

, 0, . . . , 0).

Proof For (1, n)- and (n, 1)-flips this follows from Proposition 11.3 and Lemma 6.1,since (1, n)-flip is just the connected sumwith an elementary multi-fan, and (n, 1)-flipis its inverse.

Now we consider the remaining cases. (p, q)-flips with p = 1 and q = 1 donot change the vertex set. Let [m] denote the vertex set of K and K ′, and S ⊂ [m]denote the set of vertices at which the flip is performed. We have |S| = n + 1 andK |S ∼= ∂Δp−1 ∗ Δq−1 and K ′|S ∼= Δp−1 ∗ ∂Δq−1. Let [p] be the set of vertices ofΔp−1. Let I[m]\S denote the ideal in R[∂1, . . . , ∂m] generated by ∂i , (i ∈ [m]\S).

Claim 11.7 I[m]\S ∩ Ann VΔ = I[m]\S ∩ Ann VΔ′ .

Proof In the group ofmulti-fans with a given characteristic functionwe have a relationΔ′ = Δ+T , where T is an elementary multi-fan based on the vertex set S. Informally,to perform a flip on a multi-fan is the same as “to add a boundary of a simplex”, whichcancels the cones from ∂Δp−1 ∗ Δq−1 and adds the cones from Δp−1 ∗ ∂Δq−1.Therefore VΔ′ = VΔ + VT , where VT is the polynomial which essentially dependsonly on the variables ci , i ∈ S. If D ∈ Ann VΔ ∩ I[m]\S then D annihilates both VΔ

and VT . Thus it annihilates VΔ′ = VΔ + VT and the claim follows. ��

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We have a diagram of inclusions of graded ideals in R[∂1, . . . , ∂m]:

Ann VΔ� �

��

I[m]\S ∩ Ann VΔ = I[m]\S ∩ Ann VΔ′� ��

��

Ann VΔ′� �

��I[m]\S + Ann VΔ I[m]\S� �� I[m]\S + Ann VΔ′

It follows that the quotients of the vertical inclusions are isomorphic as gradedvector spaces. Therefore

dm(Δ′) − dm(Δ) = dm(R[m]/Ann VΔ′) − dm(R[m]/Ann VΔ)

= dm(R[m]/(I[m]\S + Ann VΔ′))

−dm(R[m]/(I[m]\S + Ann VΔ)).

Since I[m]\S is the ideal generated by ∂i , (i /∈ S), the ring R[m]/(I[m]\S + Ann VΔ)

coincides with some quotient ring B of the polynomials in variables ∂i , (i ∈ S), thatis B = R[S]/Rels. The linear relations θ j = ∑

i∈[m] λi, j∂i in Ann VΔ induce therelations

∑i∈S λi, j∂i in Rels. Since the values of λ on S are in general position,

these induced relations are linearly independent. We have n linear relations on n + 1variables, thus all variables are expressed through a single variable t , and we haveB ∼= R[t]/J . Since we are in a graded situation, and B is a finite dimensional algebra,J is a principal ideal generated by t p for some p � 0. Hence

dm(R[m]/(I[m]\S + Ann VΔ)) = dmB = (1, . . . , 1︸︷︷︸p

, 0, . . . , 0).

Now notice that there is a Stanley–Reisner relation∏

i∈[p] ∂i = 0 corresponding to

non-simplex [p] in K (recall that [p] is the set of vertices of ∂Δp−1 inside ∂Δp−1 ∗Δq−1 ⊂ K ). Therefore we have a relation t p = 0 in B. This implies p � p.

Applying the same arguments to Δ′, we get

dm(R[m]/(I[m]\S + Ann VΔ′)) = (1, . . . , 1︸︷︷︸q

, 0, . . . , 0),

where q � q. Hence we have

dm(Δ′) − dm(Δ) = (1, . . . , 1︸︷︷︸q

, 0, . . . , 0) − (1, . . . , 1︸︷︷︸p

, 0, . . . , 0).

Note that the vector on the left hand side is symmetric. Hence the vector on the righthand side is symmetric. If at least one inequality p � p or q � q is strict, the vectorat the right is not symmetric. Thus p = p, q = q, and the statement follows. ��

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12 Cohomology of Torus Manifolds

12.1 Multi-Fans of Torus Manifolds

Recall that a torus manifold X is an oriented closed manifold of dimension 2n with aneffective action of n-dimensional compact torus T having at least one fixed point, andprescribed orientations of characteristic submanifolds. Any torus manifold determinesa non-singular multi-fan in the Lie algebra L(T ) ∼= R

n of the torus as follows (seedetails in Hattori and Masuda 2003).

Let Xi , i ∈ [m] be the characteristic submanifolds. Let M be a connected compo-nent of a non-empty intersection Xi1 ∩· · ·∩Xik for some k > 0 and {i1, . . . , ik} ⊂ [m],and assume that M has at least one fixed point. Such submanifold will be called aface submanifold. We also assume that the manifold X itself is a face submanifoldcorresponding to k = 0. It easily follows from the transversality of characteristic sub-manifolds thatM has codimension 2k. LetΣX be a poset of all face submanifolds of Xordered by reversed inclusion. The basic representation theory of a torus implies thatΣX is a pure simplicial poset of dimension n − 1 on the vertex set [m]. The maximalsimplices of ΣX correspond to the fixed points of X .

Given orientations of X and Xi , i ∈ [m], each fixed point obtains a sign. Thisdetermines a sign function σX : Σ

〈n〉X → {−1,+1}.

Finally, let Ti denote a circle subgroup fixing Xi , for i ∈ [m]. The orientation of Xi

determines the orientation of the 2-dimensional normal bundle of Xi , which in turndetermines an orientation of Ti . Therefore we have a well-defined primitive element

λX (i) ∈ Hom(S1, T n) ∼= Zn ⊂ R

n ∼= L(T n).

This gives a characteristic function λX : [m] → Rn . These constructions determine a

multi-fan ΔX := (ΣX , σX , λX ) associated with a torus manifold X . This multi-fan isnon-singular and complete (Hattori and Masuda 2003).

As described in Sect. 2.2, we may turn the data “simplicial poset + sign function”into the data “simplicial complex + weight function”. Let KX and wX denote thesimplicial complex and the weight function corresponding to ΔX .

In the following we assume that each non-empty intersection of characteristic sub-manifolds is connected and contains at least one fixed point. This assumption implies,in particular, that ΣX is a simplicial complex, and therefore KX = ΣX and the weightfunction wX coincides with σX .

12.2 Face Subalgebra in Cohomology

Let X be a torus manifold and ΔX be the corresponding multi-fan. Let F∗(X) ⊂H∗(X; R) be the vector subspace spanned by the cohomology classes Poincare dualto face submanifolds. Since the intersection of two face submanifolds is either a facesubmanifold or empty, F∗(X) is a subalgebra. This subalgebra is multiplicativelygenerated by the classes of characteristic submanifolds.

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The simplices of KX = ΣX correspond to face submanifolds, and there exists a ringhomomorphism R[KX ] → H∗

T (X) defined as follows: the element xI = ∏i∈I xi ∈

R[KX ]2|I | corresponding to the simplex I ∈ KX maps to the equivariant cohomologyclass dual to the face submanifold XI = ⋂

i∈I Xi . There is a natural homomorphismH∗T (X) → H∗(X) induced by the inclusion of a fiber in the Borel fibration X ↪→

X ×T ETπ→ BT . We have a commutative square of algebra homomorphisms

H∗T (ΔX ) −−−−→ H∗(ΔX )⏐⏐2 ⏐⏐2H∗T (X) −−−−→ H∗(X).

(12.1)

Recall from the end of Sect. 4.2 that H∗T (ΔX ) denotes the Stanley–Reisner ring of the

underlying simplicial complex KX , H∗(ΔX ) is its quotient by the linear system ofparameters, so the upper horizontal arrow in the diagram (12.1) is the natural quotienthomomorphism. By definition, F∗(X) is the image of the right vertical map. Hencewe have an epimorphism of algebras H∗(ΔX ) � F∗(X).

Theorem 12.1 There exists a well-defined epimorphism of algebras F∗(X) �A∗(ΔX ).

Proof The epimorphism H∗(ΔX ) → F∗(X) is compatible with the integration maps∫ΔX

: H2n(ΔX ) → R (the multi-fan integration) and∫X : F2n(X) → R (integration

over the manifold X ), see Hattori and Masuda (2003). Lemma 7.4 implies that theinduced map PD(H∗(ΔX ),

∫ΔX

) → PD(F∗(X),∫X ) is an isomorphism. Thus we

have a natural epimorphism F∗(X) � PD(F∗(X),∫X ) ∼= PD(H∗(ΔX ),

∫ΔX

) =A∗(ΔX ). ��

Therefore the part of the cohomology ring generated by characteristic submanifoldsis clamped between two algebras defined combinatorially:

H∗(ΔX ) �� F∗(X) ��

��

A∗(ΔX )

H∗(X)

(12.2)

Corollary 12.2 Betti numbers of a torus manifold X are bounded below by the dimen-sions of graded components of A∗(ΔX ).

Remark 12.3 For complete smooth toric varieties and for quasitoric manifolds allarrows in the diagram above are isomorphisms as follows from Danilov–JurkiewiczandDavis–Januszkiewicz (Davis and Januszkiewicz 1991) theorems respectively. Thisfact was later generalized to simplicial posets by Masuda and Panov (2006). If KX isa sphere, both horizontal maps are isomorphisms, so the face part of cohomology iscompletely determined by a multi-fan, while the vertical map may be non-trivial. Asan example in which such phenomena occur, take an equivariant connected sum of a

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(quasi)toric manifold with a manifold on which the torus acts freely and whose orbitspace is not a homology sphere. Finally, there exist many examples of torus manifoldsfor which all arrows in (12.2) are nontrivial (Ayzenberg 2016).

Recall that H∗(ΔX ) is linearly generated by the square-free monomials xI cor-responding to simplices I ∈ K . There may exist certain linear relations on theseelements coming from Minkowski relations:

∑I :|I |=k a(I )λ(I )μxI , where μ ∈ ΛkV

and a is a function on (k − 1)-simplices of K .

Conjecture 12.4 As a vector space, F2k(X) is generated by the elements {xI }I∈Ksubject to the Minkowski relations

∑I :|I |=k a(I )λ(I )μxI = 0, where μ runs over

ΛkV and a runs over all functions such that the element

∑I :|I |=k

a(I )[XI /T ]

bounds in Cn−k(X/T ; R).

This question is closely related to Problem 9.6. It can be seen that whenever∑I :|I |=k a(I )[XI /T ] bounds in Cn−k(X/T ; R), the element

∑I :|I |=k a(I )FI is a

cycle of multi-polytopes, therefore∑

I :|I |=k a(I )λ(I )μxI vanishes inA∗(ΔX ). How-ever, there may be cycles of multi-polytopes such that the corresponding elements∑

I :|I |=k a(I )[XI /T ] do not bound in the orbit space. This observation represents thefact that the right arrow in (12.2) can be nontrivial.

If X is an oriented manifold with locally standard torus action, having trivial freepart and acyclic proper faces of the orbit space, the conjecture is proved in Ayzen-berg (2016). Informally, this situation corresponds to the case when the underlyingsimplicial complex of ΔX is an oriented homology manifold.

References

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